problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 8
values | synthetic bool 1
class | __index_level_0__ int64 0 742k |
|---|---|---|---|---|---|---|---|---|---|
## Task Condition
Derive the equation of the normal to the given curve at the point with abscissa $x_{0}$.
$y=\sqrt[3]{x^{2}}-20, x_{0}=-8$ | ## Solution
Let's find $y^{\prime}:$
$$
y^{\prime}=\left(\sqrt[3]{x^{2}}-20\right)^{\prime}=\left(x^{\frac{2}{3}-20}\right)^{\prime}=\frac{2}{3} \cdot x^{-\frac{1}{3}}=\frac{2}{3 \sqrt[3]{x}}
$$
Then:
$y_{0}^{\prime}=y^{\prime}\left(x_{0}\right)=\frac{2}{3 \sqrt[3]{x_{0}}}=\frac{2}{3 \sqrt[3]{-8}}=\frac{2}{3 \cdot(... | 3x+8 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,798 |
## Problem Statement
Find the differential $d y$.
$y=x \ln \left|x+\sqrt{x^{2}+3}\right|-\sqrt{x^{2}+3}$ | ## Solution
$$
\begin{aligned}
& d y=y^{\prime} \cdot d x=\left(x \ln \left|x+\sqrt{x^{2}+3}\right|-\sqrt{x^{2}+3}\right)^{\prime} d x= \\
& =\left(\ln \left|x+\sqrt{x^{2}+3}\right|+x \cdot \frac{1}{x+\sqrt{x^{2}+3}} \cdot\left(1+\frac{1}{2 \sqrt{x^{2}+3}} \cdot 2 x\right)-\frac{1}{2 \sqrt{x^{2}+3}} \cdot 2 x\right) d... | \ln|x+\sqrt{x^{2}+3}|\cdot | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,799 |
## Task Condition
Approximately calculate using the differential.
$y=\sqrt[3]{x^{2}+2 x+5}, x=0.97$ | ## Solution
If the increment $\Delta x = x - x_{0 \text{ of the argument }} x$ is small in absolute value, then
$$
f(x) = f\left(x_{0} + \Delta x\right) \approx f\left(x_{0}\right) + f^{\prime}\left(x_{0}\right) \cdot \Delta x
$$
Choose:
$x_{0} = 1$
Then:
$\Delta x = -0.03$
Calculate:
$$
\begin{aligned}
& y(1) =... | 1.99 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,800 |
## Task Condition
Find the derivative.
$y=\frac{x^{2}}{2 \sqrt{1-3 x^{4}}}$ | ## Solution
$y^{\prime}=\left(\frac{x^{2}}{2 \sqrt{1-3 x^{4}}}\right)^{\prime}=\frac{2 x \cdot \sqrt{1-3 x^{4}}-x^{2} \cdot \frac{1}{2 \sqrt{1-3 x^{4}}} \cdot\left(-12 x^{3}\right)}{2\left(1-3 x^{4}\right)}=$
$=\frac{x \cdot \sqrt{1-3 x^{4}}+\frac{3 x^{5}}{\sqrt{1-3 x^{4}}}}{1-3 x^{4}}=\frac{x \cdot\left(1-3 x^{4}\ri... | \frac{x}{(1-3x^{4})\sqrt{1-3x^{4}}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,801 |
## Task Condition
Find the derivative.
$$
y=\frac{2}{3} \sqrt{\left(\operatorname{arctan} e^{x}\right)^{3}}
$$ | ## Solution
$y^{\prime}=\left(\frac{2}{3} \sqrt{\left(\operatorname{arctg} e^{x}\right)^{3}}\right)^{\prime}=\frac{2}{3} \cdot \frac{3}{2} \cdot \sqrt{\operatorname{arctg} e^{x}} \cdot \frac{1}{1+e^{2 x}} \cdot e^{x}=\frac{e^{x} \cdot \sqrt{\operatorname{arctg} e^{x}}}{1+e^{2 x}}$
## Problem Kuznetsov Differentiation... | \frac{e^{x}\cdot\sqrt{\operatorname{arctg}e^{x}}}{1+e^{2x}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,802 |
## Problem Statement
Find the derivative.
$y=\ln \frac{a^{2}+x^{2}}{a^{2}-x^{2}}$ | ## Solution
$$
\begin{aligned}
& y^{\prime}=\left(\ln \frac{a^{2}+x^{2}}{a^{2}-x^{2}}\right)^{\prime}=\frac{a^{2}-x^{2}}{a^{2}+x^{2}} \cdot \frac{2 x\left(a^{2}-x^{2}\right)-\left(a^{2}+x^{2}\right) \cdot(-2 x)}{\left(a^{2}-x^{2}\right)^{2}}= \\
& =\frac{2 x\left(a^{2}-x^{2}+a^{2}+x^{2}\right)}{a^{4}-x^{4}}=\frac{4 a^... | \frac{4^{2}x}{^{4}-x^{4}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,803 |
## Task Condition
Find the derivative.
$$
y=\frac{\sin (\cos 3) \cdot \cos ^{2} 2 x}{4 \sin 4 x}
$$ | ## Solution
$$
\begin{aligned}
& y^{\prime}=\left(\frac{\sin (\cos 3) \cdot \cos ^{2} 2 x}{4 \sin 4 x}\right)^{\prime}=\sin (\cos 3) \cdot\left(\frac{\cos ^{2} 2 x}{8 \sin 2 x \cdot \cos 2 x}\right)^{\prime}= \\
& =\sin (\cos 3) \cdot\left(\frac{\cos 2 x}{8 \sin 2 x}\right)^{\prime}=\frac{\sin (\cos 3)}{8} \cdot(\oper... | -\frac{\sin(\cos3)}{4\sin^{2}2x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,804 |
## Task Condition
Find the derivative.
$y=\sqrt{\frac{2}{3}} \cdot \operatorname{arctg} \frac{3 x-1}{\sqrt{6 x}}$ | ## Solution
$$
\begin{aligned}
& y^{\prime}=\left(\sqrt{\frac{2}{3}} \cdot \operatorname{arctg} \frac{3 x-1}{\sqrt{6 x}}\right)^{\prime}=\sqrt{\frac{2}{3}} \cdot \frac{1}{1+\left(\frac{3 x-1}{\sqrt{6 x}}\right)^{2}}\left(\frac{3 x-1}{\sqrt{6 x}}\right)^{\prime}= \\
& =\sqrt{\frac{2}{3}} \cdot \frac{6 x}{6 x+(3 x-1)^{2... | \frac{3x+1}{\sqrt{x}(9x^{2}+1)} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,805 |
## Task Condition
Find the derivative.
$y=-\frac{1}{2} \cdot \ln \left(\tanh \frac{x}{2}\right)-\frac{\cosh x}{2 \sinh^{2} x}$ | ## Solution
$$
\begin{aligned}
& y^{\prime}=\left(-\frac{1}{2} \cdot \ln \left(\operatorname{th} \frac{x}{2}\right)-\frac{\operatorname{ch} x}{2 \operatorname{sh}^{2} x}\right)^{\prime}= \\
& =-\frac{1}{2} \cdot \frac{1}{\operatorname{th} \frac{x}{2}} \cdot\left(\operatorname{th} \frac{x}{2}\right)^{\prime}-\frac{\ope... | \frac{1}{\sinh^3x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,806 |
## Task Condition
Find the derivative.
$y=x^{\arcsin x}$ | ## Solution
$y=x^{\arcsin x}$
$\ln y=\arcsin x \cdot \ln x$
$\frac{y^{\prime}}{y}=(\arcsin x \cdot \ln x)^{\prime}=\frac{1}{\sqrt{1-x^{2}}} \cdot \ln x+\arcsin x \cdot \frac{1}{x}=$
$=\frac{\ln x}{\sqrt{1-x^{2}}}+\frac{\arcsin x}{x}$
$y^{\prime}=y \cdot\left(\frac{\ln x}{\sqrt{1-x^{2}}}+\frac{\arcsin x}{x}\right)=... | x^{\arcsinx}\cdot(\frac{\lnx}{\sqrt{1-x^{2}}}+\frac{\arcsinx}{x}) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,807 |
## Problem Statement
Find the derivative.
$y=\frac{x^{4}}{81} \cdot \arcsin \frac{3}{x}+\frac{1}{81}\left(x^{2}+18\right) \sqrt{x^{2}-9}, x>0$ | ## Solution
$$
\begin{aligned}
& y^{\prime}=\left(\frac{x^{2}}{81} \cdot \arcsin \frac{3}{x}+\frac{1}{81}\left(x^{2}+18\right) \sqrt{x^{2}-9}\right)^{\prime}= \\
& =\frac{2 x}{81} \cdot \arcsin \frac{3}{x}+\frac{x^{2}}{81} \cdot \frac{1}{\sqrt{1-\frac{9}{x^{2}}}} \cdot\left(-\frac{3}{x^{2}}\right)+ \\
& +\frac{1}{81} ... | \frac{2x}{81}\cdot\arcsin\frac{3}{x}+\frac{x\cdot(x^{2}-1)}{27\sqrt{x^{2}-9}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,808 |
## Task Condition
Find the derivative.
$$
y=\sqrt{1+x^{2}} \operatorname{arctg} x-\ln \left(x+\sqrt{1+x^{2}}\right)
$$ | ## Solution
$$
\begin{aligned}
& y^{\prime}=\left(\sqrt{1+x^{2}} \arctan x-\ln \left(x+\sqrt{1+x^{2}}\right)\right)^{\prime}= \\
& =\frac{1}{2 \sqrt{1+x^{2}}} \cdot 2 x \cdot \arctan x+\sqrt{1+x^{2}} \cdot \frac{1}{1+x^{2}}-\frac{1}{\left(x+\sqrt{1+x^{2}}\right)} \cdot\left(1+\frac{1}{2 \sqrt{1+x^{2}}} \cdot 2 x\right... | \frac{x\cdot\arctanx}{\sqrt{1+x^{2}}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,809 |
## Problem Statement
Find the derivative.
$$
y=\left(a^{2}+b^{2}\right)^{-\frac{1}{2}} \cdot \arcsin \left(\frac{\sqrt{a^{2}+b^{2}} \cdot \sin x}{b}\right)
$$ | ## Solution
$$
\begin{aligned}
& y^{\prime}=\left(\left(a^{2}+b^{2}\right)^{-\frac{1}{2}} \cdot \arcsin \left(\frac{\sqrt{a^{2}+b^{2}} \cdot \sin x}{b}\right)\right)^{\prime}= \\
& =\left(a^{2}+b^{2}\right)^{-\frac{1}{2}} \cdot \frac{1}{\sqrt{1-\left(\frac{\sqrt{a^{2}+b^{2} \cdot \sin x}}{b}\right)^{2}}} \cdot\left(\f... | \frac{\cosx}{\sqrt{b^{2}\cdot\cos^{2}x-^{2}\cdot\sin^{2}x}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,810 |
## Task Condition
Find the derivative $y_{x}^{\prime}$.
$$
\left\{\begin{array}{l}
x=\sqrt{2 t-t^{2}} \\
y=\arcsin (t-1)
\end{array}\right.
$$ | ## Solution
$x_{t}^{\prime}=\left(\sqrt{2 t-t^{2}}\right)^{\prime}=\frac{1}{2 \sqrt{2 t-t^{2}}} \cdot(2-2 t)=\frac{1-t}{\sqrt{2 t-t^{2}}}$
$y_{t}^{\prime}=(\arcsin (t-1))^{\prime}=\frac{1}{\sqrt{1-(t-1)^{2}}}=\frac{1}{\sqrt{1-t^{2}-2 t-1}}=\frac{1}{\sqrt{-t^{2}-2 t}}$
We obtain:
$$
\begin{aligned}
& y_{x}^{\prime}=... | \frac{1}{(1-)}\cdot\sqrt{\frac{-2}{+2}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,811 |
## Problem Statement
Derive the equations of the tangent and normal lines to the curve at the point corresponding to the parameter value $t=t_{0}$.
\[
\left\{\begin{array}{l}
x=\arcsin \frac{t}{\sqrt{1+t^{2}}} \\
y=\arccos \frac{1}{\sqrt{1+t^{2}}}
\end{array}\right.
\]
$t_{0}=-1$ | ## Solution
Since $t_{0}=-1$, then
$x_{0}=\arcsin \frac{-1}{\sqrt{1+(-1)^{2}}}=\arcsin \frac{-1}{\sqrt{2}}=-\frac{\pi}{4}$
$y_{0}=\arccos \frac{1}{\sqrt{1+(-1)^{2}}}=\arccos \frac{1}{\sqrt{2}}=\frac{\pi}{4}$
Find the derivatives:
$$
\begin{aligned}
& x_{t}^{\prime}=\left(\arcsin \frac{t}{\sqrt{1+t^{2}}}\right)^{\pr... | 2x+\frac{3\pi}{4} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,812 |
Condition of the problem
Find the $n$-th order derivative.
$y=a^{3 x}$ | ## Solution
$y=a^{3 x}=\left(e^{\ln a}\right)^{3 x}=e^{3 x \cdot \ln a}$
$y^{\prime}=\left(e^{3 x \cdot \ln a}\right)^{\prime}=e^{3 x \cdot \ln a} \cdot 3 \ln a$
$y^{\prime \prime}=\left(y^{\prime}\right)^{\prime}=\left(e^{3 x \cdot \ln a} \cdot 3 \ln a\right)^{\prime}=e^{3 x \cdot \ln a} \cdot 3^{2} \ln ^{2} a$
$\c... | y^{(n)}=^{3x}\cdot3^{n}\ln^{n} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,813 |
## Task Condition
Find the derivative of the specified order.
$y=\left(4 x^{3}+5\right) e^{2 x+1}, y^{V}=?$ | ## Solution
$y^{\prime}=\left(\left(4 x^{3}+5\right) e^{2 x+1}\right)^{\prime}=12 x^{2} \cdot e^{2 x+1}+2\left(4 x^{3}+5\right) e^{2 x+1}=$
$=2\left(4 x^{3}+6 x^{2}+5\right) e^{2 x+1}$
$y^{\prime \prime}=\left(y^{\prime}\right)^{\prime}=\left(2\left(4 x^{3}+6 x^{2}+5\right) e^{2 x+1}\right)^{\prime}=$
$=2\left(12 x... | 32(4x^{3}+30x^{2}+60x+35)e^{2x+1} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,814 |
## Problem Statement
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$\left\{\begin{array}{l}x=\frac{1}{t} \\ y=\frac{1}{1+t^{2}}\end{array}\right.$ | ## Solution
$x_{t}^{\prime}=\left(\frac{1}{t}\right)^{\prime}=-\frac{1}{t^{2}}$
$y_{t}^{\prime}=\left(\frac{1}{1+t^{2}}\right)^{\prime}=-\frac{1}{\left(1+t^{2}\right)^{2}} \cdot 2 t=-\frac{2 t}{\left(1+t^{2}\right)^{2}}$
We obtain:
$y_{x}^{\prime}=\frac{y_{t}^{\prime}}{x_{t}^{\prime}}=\frac{-\frac{2 t}{\left(1+t^{2... | \frac{2(^{2}-3)\cdot^{4}}{(1+^{2})^{3}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,815 |
## Task Condition
Based on the definition of the derivative, find $f^{\prime}(0)$:
$$
f(x)=\left\{\begin{array}{c}
\operatorname{arctg} x \cdot \sin \frac{7}{x}, x \neq 0 \\
0, x=0
\end{array}\right.
$$ | ## Solution
By definition, the derivative at the point $x=0$:
$f^{\prime}(0)=\lim _{\Delta x \rightarrow 0} \frac{f(0+\Delta x)-f(0)}{\Delta x}$
Based on the definition, we find:
$$
\begin{aligned}
& f^{\prime}(0)=\lim _{\Delta x \rightarrow 0} \frac{f(0+\Delta x)-f(0)}{\Delta x}=\lim _{\Delta x \rightarrow 0} \fra... | f^{\}(0) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,817 |
Condition of the problem
To derive the equation of the tangent line to the given curve at the point with abscissa $x_{0}$.
$y=2 x^{2}+3, x_{0}=-1$ | ## Solution
Let's find $y^{\prime}$:
$y^{\prime}=\left(2 x^{2}+3\right)^{\prime}=4 x$
Then:
$y_{0}^{\prime}=y^{\prime}\left(x_{0}\right)=4 x_{0}=4 \cdot(-1)=-4$
Since the function $y^{\prime}$ at the point $x_{0}$ has a finite derivative, the equation of the tangent line is:
$$
\begin{aligned}
& y-y_{0}=y_{0}^{\pr... | -4x+1 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,818 |
## Condition of the problem
Find the differential $d y$
$$
y=x \sqrt{4-x^{2}}+a \cdot \arcsin \frac{x}{2}
$$ | ## Solution
$$
\begin{aligned}
& d y=y^{\prime} \cdot d x=\left(x \sqrt{4-x^{2}}+a \cdot \arcsin \frac{x}{2}\right)^{\prime} d x= \\
& =\left(\sqrt{4-x^{2}}+x \cdot \frac{1}{2 \sqrt{4-x^{2}}} \cdot(-2 x)+a \cdot \frac{1}{2 \sqrt{1-\left(\frac{x}{2}\right)^{2}}} \cdot \frac{1}{2}\right) d x= \\
& =\left(\frac{4-x^{2}}{... | \frac{-2x^{2}+4+2}{\sqrt{4-x^{2}}}\cdot | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,819 |
## Task Condition
Approximately calculate using the differential.
$y=x^{6}, x=2.01$ | ## Solution
If the increment $\Delta x = x - x_{0}$ of the argument $x$ is small in absolute value, then
$f(x) = f\left(x_{0} + \Delta x\right) \approx f\left(x_{0}\right) + f^{\prime}\left(x_{0}\right) \cdot \Delta x$
Choose:
$x_{0} = 2$
Then:
$\Delta x = 0.01$
Calculate:
$y(2) = 2^{6} = 64$
$y^{\prime} = \le... | 65.92 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,820 |
## Task Condition
Find the derivative.
$y=\frac{1+x^{2}}{2 \sqrt{1+2 x^{2}}}$ | ## Solution
$$
\begin{aligned}
& y^{\prime}=\left(\frac{1+x^{2}}{2 \sqrt{1+2 x^{2}}}\right)^{\prime}=\frac{2 x \cdot \sqrt{1+2 x^{2}}-\left(1+x^{2}\right) \cdot \frac{1}{2 \sqrt{1+2 x^{2}}} \cdot 4 x}{2\left(1+2 x^{2}\right)}= \\
& =\frac{2 x \cdot\left(1+2 x^{2}\right)-\left(1+x^{2}\right) \cdot 2 x}{2\left(1+2 x^{2}... | \frac{x^{3}}{(1+2x^{2})\sqrt{1+2x^{2}}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,821 |
## Task Condition
Find the derivative.
$$
y=e^{a x}\left(\frac{1}{2 a}+\frac{a \cdot \cos 2 b x+2 b \cdot \sin 2 b x}{2\left(a^{2}+4 b^{2}\right)}\right)
$$ | ## Solution
$y^{\prime}=\left(e^{a x}\left(\frac{1}{2 a}+\frac{a \cdot \cos 2 b x+2 b \cdot \sin 2 b x}{2\left(a^{2}+4 b^{2}\right)}\right)\right)^{\prime}=$
$=a \cdot e^{a x}\left(\frac{1}{2 a}+\frac{a \cdot \cos 2 b x+2 b \cdot \sin 2 b x}{2\left(a^{2}+4 b^{2}\right)}\right)+e^{a x} \cdot \frac{1}{2\left(a^{2}+4 b^... | e^{}\cdot\cos^{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,822 |
## Task Condition
Find the derivative.
$y=\ln \left(\sin \frac{2 x+4}{x+1}\right)$ | ## Solution
$y^{\prime}=\left(\ln \left(\sin \frac{2 x+4}{x+1}\right)\right)^{\prime}=\frac{1}{\sin \frac{2 x+4}{x+1}} \cdot\left(\sin \frac{2 x+4}{x+1}\right)^{\prime}=$
$=\frac{1}{\sin \frac{2 x+4}{x+1}} \cdot \cos \frac{2 x+4}{x+1} \cdot\left(\frac{2 x+4}{x+1}\right)^{\prime}=$
$=\operatorname{ctg} \frac{2 x+4}{x... | -\frac{2}{(x+1)^{2}}\cdot\operatorname{ctg}\frac{2x+4}{x+1} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,823 |
## Task Condition
Find the derivative.
$y=8 \sin (\operatorname{ctg} 3)+\frac{1}{5} \cdot \frac{\sin ^{2} 5 x}{\cos 10 x}$ | Solution
$y^{\prime}=\left(8 \sin (\operatorname{ctg} 3)+\frac{1}{5} \cdot \frac{\sin ^{2} 5 x}{\cos 10 x}\right)^{\prime}=0+\frac{1}{5} \cdot\left(\frac{\sin ^{2} 5 x}{\cos 10 x}\right)^{\prime}=$
$=\frac{1}{5} \cdot \frac{2 \sin 5 x \cdot \cos 5 x \cdot 5 \cdot \cos 10 x-\sin ^{2} 5 x \cdot(-\sin 10 x) \cdot 10}{\c... | \frac{\operatorname{tg}10x}{\cos10x} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,824 |
## Problem Statement
Find the derivative.
$y=\frac{4+x^{4}}{x^{3}} \cdot \operatorname{arctg} \frac{x^{2}}{2}+\frac{4}{x}$ | ## Solution
$y^{\prime}=\left(\frac{4+x^{4}}{x^{3}} \cdot \operatorname{arctg} \frac{x^{2}}{2}+\frac{4}{x}\right)^{\prime}=$
$=\frac{4 x^{3} \cdot x^{3}-\left(4+x^{4}\right) \cdot 3 x^{2}}{x^{6}} \cdot \operatorname{arctg} \frac{x^{2}}{2}+\frac{4+x^{4}}{x^{3}} \cdot \frac{1}{1+\left(\frac{x^{2}}{2}\right)^{2}} \cdot ... | \frac{x^{4}-12}{x^{4}}\cdot\operatorname{arctg}\frac{x^{2}}{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,825 |
Condition of the problem
Find the derivative.
$y=\frac{\operatorname{ch} x}{\sqrt{\operatorname{sh} 2 x}}$ | ## Solution
$$
\begin{aligned}
& y^{\prime}=\left(\frac{\cosh x}{\sqrt{\sinh 2 x}}\right)^{\prime}=\left(\frac{\cosh x}{\sqrt{2 \sinh x \cdot \cosh x}}\right)^{\prime}=\left(\sqrt{\frac{\cosh x}{2 \sinh x}}\right)^{\prime}= \\
& =\frac{1}{\sqrt{2}} \cdot\left(\sqrt{\frac{\cosh x}{\sinh x}}\right)^{\prime}=\frac{1}{\sq... | -\frac{1}{2\sinhx\sqrt{\sinh2x}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,826 |
## Task Condition
Find the derivative.
$y=\left(x^{3}+4\right)^{\tan x}$ | ## Solution
$y=\left(x^{3}+4\right)^{\tan x}$
$\ln y=\ln \left(x^{3}+4\right)^{\tan x}=\tan x \cdot \ln \left(x^{3}+4\right)$
$\frac{y'}{y}=\frac{1}{1+x^{2}} \cdot \ln \left(x^{3}+4\right)+\tan x \cdot \frac{1}{x^{3}+4} \cdot 3 x^{2}=$
$=\frac{\ln \left(x^{3}+4\right)}{1+x^{2}}+\frac{3 x^{2} \cdot \tan x}{x^{3}+4}$... | (x^{3}+4)^{\tanx}\cdot(\frac{\ln(x^{3}+4)}{1+x^{2}}+\frac{3x^{2}\cdot\tanx}{x^{3}+4}) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,827 |
## Task Condition
Find the derivative.
$y=5 x-\ln \left(1+\sqrt{1-e^{10 x}}\right)-e^{-5 x} \cdot \arcsin \left(e^{5 x}\right)$ | ## Solution
$$
\begin{aligned}
& y=\left(5 x-\ln \left(1+\sqrt{1-e^{10 x}}\right)-e^{-5 x} \cdot \arcsin \left(e^{5 x}\right)\right)^{\prime}= \\
& =5-\frac{1}{1+\sqrt{1-e^{10 x}}} \cdot \frac{1}{2 \sqrt{1-e^{10 x}}} \cdot\left(-10 e^{10 x}\right)-
\end{aligned}
$$
$$
\begin{aligned}
& -\left(-5 e^{-5 x} \cdot \arcsi... | 5e^{-5x}\cdot\arcsin(e^{5x}) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,828 |
## Problem Statement
Find the derivative.
$$
y=\frac{1}{12} \cdot \ln \frac{x^{4}-x^{2}+1}{\left(x^{2}+1\right)^{2}}-\frac{1}{2 \sqrt{3}} \operatorname{arctg} \frac{\sqrt{3}}{2 x^{2}-1}
$$ | ## Solution
$$
\begin{aligned}
& y^{\prime}=\left(\frac{1}{12} \cdot \ln \frac{x^{4}-x^{2}+1}{\left(x^{2}+1\right)^{2}}-\frac{1}{2 \sqrt{3}} \operatorname{arctg} \frac{\sqrt{3}}{2 x^{2}-1}\right)^{\prime}= \\
& =\frac{1}{12} \cdot \frac{\left(x^{2}+1\right)^{2}}{x^{4}-x^{2}+1} \cdot \frac{\left(4 x^{3}-2 x\right) \cdo... | \frac{x^{3}}{(x^{4}-x^{2}+1)\cdot(x^{2}+1)} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,829 |
## Task Condition
Find the derivative.
$$
y=\frac{1}{2 \sin \frac{\alpha}{2}} \cdot \operatorname{arctg} \frac{2 x \sin \frac{\alpha}{2}}{1-x^{2}}
$$ | ## Solution
$$
\begin{aligned}
& y^{\prime}=\left(\frac{1}{2 \sin \frac{\alpha}{2}} \cdot \operatorname{arctg} \frac{2 x \sin \frac{\alpha}{2}}{1-x^{2}}\right)^{\prime}= \\
& =\frac{1}{2 \sin \frac{\alpha}{2}} \cdot \frac{1}{1+\left(\frac{2 x \sin \frac{\alpha}{2}}{1-x^{2}}\right)^{2}} \cdot \frac{2 \sin \frac{\alpha}... | \frac{1+x^{2}}{(1-x^{2})^{2}+4x^{2}\cdot\sin^{2}\frac{\alpha}{2}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,830 |
## Problem Statement
Find the derivative $y_{x}^{\prime}$.
$$
\left\{\begin{array}{l}
x=\arcsin \left(\sqrt{1-t^{2}}\right) \\
y=(\arccos t)^{2}
\end{array}\right.
$$ | ## Solution
$x_{t}^{\prime}=\left(\arcsin \left(\sqrt{1-t^{2}}\right)\right)^{\prime}=\frac{1}{\sqrt{1-\left(\sqrt{1-t^{2}}\right)^{2}}} \cdot \frac{1}{2 \sqrt{1-t^{2}}} \cdot(-2 t)=$
$=-\frac{1}{\sqrt{1-1+t^{2}}} \cdot \frac{t}{\sqrt{1-t^{2}}}=-\frac{1}{t} \cdot \frac{t}{\sqrt{1-t^{2}}}=-\frac{1}{\sqrt{1-t^{2}}}$
$... | 2\arccos | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,831 |
## Problem Statement
Derive the equations of the tangent and normal lines to the curve at the point corresponding to the parameter value $t=t_{0}$.
\[
\left\{\begin{array}{l}
x=\arcsin \frac{t}{\sqrt{1+t^{2}}} \\
y=\arccos \frac{1}{\sqrt{1+t^{2}}}
\end{array}\right.
\]
$t_{0}=1$ | ## Solution
Since $t_{0}=1$, then
$x_{0}=\arcsin \frac{1}{\sqrt{1+1^{2}}}=\arcsin \frac{1}{\sqrt{2}}=\frac{\pi}{4}$
$y_{0}=\arccos \frac{1}{\sqrt{1+1^{2}}}=\arccos \frac{1}{\sqrt{2}}=\frac{\pi}{4}$
Let's find the derivatives:
$x_{t}^{\prime}=\left(\arcsin \frac{t}{\sqrt{1+t^{2}}}\right)^{\prime}=\frac{1}{\sqrt{1-\l... | 2x-\frac{\pi}{4} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,832 |
## problem statement
Find the $n$-th order derivative.
$y=\sqrt[3]{e^{2 x+1}}$ | ## Solution
$$
\begin{aligned}
& y=\sqrt[3]{e^{2 x+1}}=e^{\frac{2 x+1}{3}} \\
& y^{\prime}=\left(e^{\frac{2 x+1}{3}}\right)^{\prime}=e^{\frac{2 x+1}{3}} \cdot \frac{2}{3} \\
& y^{\prime \prime}=\left(y^{\prime}\right)^{\prime}=\left(e^{\frac{2 x+1}{3}} \cdot \frac{2}{3}\right)^{\prime}=e^{\frac{2 x+1}{3}} \cdot\left(\... | (\frac{2}{3})^{n}\cdot\sqrt[3]{e^{2x+1}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,833 |
## Task Condition
Find the derivative of the specified order.
$y=e^{1-2 x} \cdot \sin (2+3 x), y^{IV}=?$ | ## Solution
$y^{\prime}=\left(e^{1-2 x} \cdot \sin (2+3 x)\right)^{\prime}=$
$=-2 e^{1-2 x} \cdot \sin (2+3 x)+3 e^{1-2 x} \cdot \cos (2+3 x)$
$y^{\prime \prime}=\left(y^{\prime}\right)=\left(-2 e^{1-2 x} \cdot \sin (2+3 x)+3 e^{1-2 x} \cdot \cos (2+3 x)\right)^{\prime}=$
$=4 e^{1-2 x} \cdot \sin (2+3 x)-6 e^{1-2 x}... | -122e^{1-2x}\cdot\sin(2+3x)-597e^{1-2x}\cdot\cos(2+3x) | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,834 |
## Task Condition
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$$
\left\{\begin{array}{l}
x=\sqrt{t^{3}-1} \\
y=\ln t
\end{array}\right.
$$ | ## Solution
$x_{t}^{\prime}=\left(\sqrt{t^{3}-1}\right)^{\prime}=\frac{1}{2 \sqrt{t^{3}-1}} \cdot 3 t^{2}=\frac{3 t^{2}}{2 \sqrt{t^{3}-1}}$
$y_{t}^{\prime}=(\ln t)^{\prime}=\frac{1}{t}$
We obtain:
$$
\begin{aligned}
& y_{x}^{\prime}=\frac{y_{t}^{\prime}}{x_{t}^{\prime}}=\left(\frac{1}{t}\right) /\left(\frac{3 t^{2}... | \frac{2(2-^{3})}{3^{6}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 47,835 |
## Problem Statement
Show that the function $y_{\text {satisfies equation (1). }}$.
\[
\begin{aligned}
& y=\frac{b+x}{1+b x} \\
& y-x \cdot y^{\prime}=b\left(1+x^{2} \cdot y^{\prime}\right)
\end{aligned}
\] | ## Solution
$y^{\prime}=\left(\frac{b+x}{1+b x}\right)^{\prime}=\frac{1 \cdot(1+b x)-(b+x) \cdot b}{(1+b x)^{2}}=$
$=\frac{1+b x-b^{2}-b x}{(1+b x)^{2}}=\frac{1-b^{2}}{(1+b x)^{2}}$
Substitute into equation (1):
$\frac{b+x}{1+b x}-x \cdot \frac{1-b^{2}}{(1+b x)^{2}}=b\left(1+x^{2} \cdot \frac{1-b^{2}}{(1+b x)^{2}}\... | proof | Algebra | proof | Yes | Yes | olympiads | false | 47,836 |
2.159. $\left(\left(\frac{a \sqrt[3]{b}}{b \sqrt{a^{3}}}\right)^{3 / 2}+\left(\frac{\sqrt{a}}{a \sqrt[8]{b^{3}}}\right)^{2}\right):\left(a^{1 / 4}+b^{1 / 4}\right)$ | ## Solution.
Domain of definition: $\left\{\begin{array}{l}a>0, \\ b>0 .\end{array}\right.$
$$
\begin{aligned}
& \left(\left(\frac{a \sqrt[3]{b}}{b \sqrt{a^{3}}}\right)^{3 / 2}+\left(\frac{\sqrt{a}}{a \sqrt[8]{b^{3}}}\right)^{2}\right):\left(a^{1 / 4}+b^{1 / 4}\right)= \\
& =\left(\frac{a^{3 / 2}(\sqrt[3]{b})^{3 / 2}... | \frac{1}{} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,838 |
2.160. $\frac{\left(a^{2} b \sqrt{b}-6 a^{5 / 3} b^{5 / 4}+12 a b \sqrt[3]{a}-8 a b^{3 / 4}\right)^{2 / 3}}{a b \sqrt[3]{a}-4 a b^{3 / 4}+4 a^{2 / 3} \sqrt{b}}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}a^{1 / 3} b^{1 / 4} \neq 2, \\ a \neq 0, \\ b \neq 0 .\end{array}\right.$
$$
\begin{aligned}
& \frac{\left(a^{2} b \sqrt{b}-6 a^{5 / 3} b^{5 / 4}+12 a b \sqrt[3]{a}-8 a b^{3 / 4}\right)^{2 / 3}}{a b \sqrt[3]{a}-4 a b^{3 / 4}+4 a^{2 / 3} \sqrt{b}}= \\
& =\frac{\l... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,839 |
2.161. $\frac{a^{3}-3 a^{2}+4+\left(a^{2}-4\right) \sqrt{a^{2}-1}}{a^{3}+3 a^{2}-4+\left(a^{2}-4\right) \sqrt{a^{2}-1}} ; a>1, a \neq \frac{2}{\sqrt{3}}$. | Solution.
$$
\begin{aligned}
& \frac{a^{3}-3 a^{2}+4+\left(a^{2}-4\right) \sqrt{a^{2}-1}}{a^{3}+3 a^{2}-4+\left(a^{2}-4\right) \sqrt{a^{2}-1}}=\frac{a^{2}(a-2)-\left(a^{2}-4\right)+\left(a^{2}-4\right) \sqrt{a^{2}-1}}{a^{2}(a+2)+\left(a^{2}-4\right)+\left(a^{2}-4\right) \sqrt{a^{2}-1}}= \\
& =\frac{(a-2)\left(a^{2}-4-... | \frac{(-2)\sqrt{+1}}{(+2)\sqrt{-1}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,840 |
2.164. $\frac{x^{2}+2 x-3+(x+1) \sqrt{x^{2}-9}}{x^{2}-2 x-3+(x-1) \sqrt{x^{2}-9}} ; x>3$. | Solution.
$$
\begin{aligned}
& \frac{x^{2}+2 x-3+(x+1) \sqrt{x^{2}-9}}{x^{2}-2 x-3+(x-1) \sqrt{x^{2}-9}}=\frac{\left(x^{2}-1\right)+2(x-1)+(x+1) \sqrt{x^{2}-9}}{\left(x^{2}-1\right)-2(x+1)+(x-1) \sqrt{x^{2}-9}}= \\
& =\frac{(x+3)(x-1)+(x+1) \sqrt{(x-3)(x+3)}}{(x-3)(x+1)+(x-1) \sqrt{(x-3)(x+3)}}= \\
& =\frac{\sqrt{x+3}... | \sqrt{\frac{x+3}{x-3}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,843 |
2.165. $\frac{t^{2}-t-6-(t+3) \sqrt{t^{2}-4}}{t^{2}+t-6-(t-3) \sqrt{t^{2}-4}} ; t>2$. | Solution.
$$
\begin{aligned}
& \frac{t^{2}-t-6-(t+3) \sqrt{t^{2}-4}}{t^{2}+t-6-(t-3) \sqrt{t^{2}-4}}=\frac{\left(t^{2}-4\right)-(t+2)-(t+3) \sqrt{t^{2}-4}}{\left(t^{2}-4\right)+(t-2)-(t-3) \sqrt{t^{2}-4}}= \\
& \left.=\frac{(t-3)(t+2)-(t+3) \sqrt{(t-2)(t+2)}}{(t+3)(t-2)-(t-3) \sqrt{(t-2)(t+2)}}=\frac{\sqrt{t+2}}{\sqrt... | -\sqrt{\frac{+2}{-2}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,844 |
2.166. $\frac{\frac{|b-1|}{b}+b \cdot|b-1|+2-\frac{2}{b}}{\sqrt{b-2+\frac{1}{b}}}$. | Solution.
Domain of definition: $b>0, b \neq 1$.
$$
\begin{aligned}
& \frac{\frac{|b-1|}{b}+b \cdot|b-1|+2-\frac{2}{b}}{\sqrt{b-2+\frac{1}{b}}}=\frac{\frac{|b-1|+b^{2} \cdot|b-1|+2 b-2}{b}}{\sqrt{\frac{b^{2}-2 b+1}{b}}}= \\
& =\frac{|b-1| \cdot\left(b^{2}+1\right)+2(b-1)}{b \sqrt{\frac{(b-1)^{2}}{b}}}=\frac{|b-1| \cd... | \frac{b^{2}-1}{\sqrt{b}}forb\in(0;1);\frac{b^{2}+3}{\sqrt{b}}forb\in(1;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,845 |
2.167. $\frac{m^{5}+m^{4} \sqrt[3]{2}+\sqrt[3]{4 m^{9}}}{\left|m^{3}-1\right|-1}$. | ## Solution.
Domain of definition: $\left\{\begin{array}{l}m \neq \sqrt[3]{2} \\ m \neq 0\end{array}\right.$
$$
\frac{m^{5}+m^{4} \sqrt[3]{2}+\sqrt[3]{4 m^{9}}}{\left|m^{3}-1\right|-1}=\frac{m^{5}+\sqrt[3]{2} \cdot m^{4}+\sqrt[3]{2^{2}} \cdot m^{3}}{\left|m^{3}-1\right|-1}=\frac{m^{3}\left(m^{2}+\sqrt[3]{2} \cdot m+\... | -(^{2}+\sqrt[3]{2}\cdot+\sqrt[3]{2^{2}})when\in(-\infty;0)\cup(0;1);\frac{^{3}}{-\sqrt[3]{2}}when\in[1;\sqrt[3]{2})\cup(\sqrt[3]{2}; | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,846 |
2.168. $\frac{x^{4}-x^{3}-x+1}{x^{3}-5 x^{2}+7 x-3} \cdot|x-3|$.
2.168. $\frac{x^{4}-x^{3}-x+1}{x^{3}-5 x^{2}+7 x-3} \cdot|x-3|$.
(Note: The mathematical expression is the same in both languages, so the translation is identical to the original text.) | Solution.
DZ: $\left\{\begin{array}{l}x \neq 1 \\ x \neq 3\end{array}\right.$
$$
\begin{aligned}
& \frac{x^{4}-x^{3}-x+1}{x^{3}-5 x^{2}+7 x-3} \cdot|x-3|=\frac{(x-1)^{2}\left(x^{2}+x+1\right)}{(x-1)^{2}(x-3)} \cdot|x-3|= \\
& =\frac{\left(x^{2}+x+1\right) \cdot|x-3|}{x-3}=\left\{\begin{array}{l}
\frac{\left(x^{2}+x+1... | -(x^{2}+x+1)whenx\in(-\infty;1)\cup(1;3);x^{2}+x+1whenx\in(3;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,847 |
2.169. $\left(\sqrt[3]{m^{2}}+n \sqrt[3]{m}+n^{2}\right) \cdot \frac{\sqrt[3]{m^{4}}-n^{3}+n^{2} \sqrt[3]{m}-m n}{m n^{-1}+n-n^{4} m^{-1}-n^{2}}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}m \neq 0, \\ n \neq 0, \\ m \neq n^{3}, \\ n^{2} \neq -m .\end{array}\right.$
$\left(\sqrt[3]{m^{2}}+n \sqrt[3]{m}+n^{2}\right) \cdot \frac{\sqrt[3]{m^{4}}-n^{3}+n^{23} \sqrt{m}-m n}{m n^{-1}+n-n^{4} m^{-1}-n^{2}}=$
$=\left(m^{2 / 3}+n m^{1 / 3}+n^{2}\right) \c... | n | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,848 |
2.170. $\frac{a^{3}+a^{2}-2 a}{a|a+2|-a^{2}+4}$. | Solution.
Domain of definition: $a \neq-2$.
$$
\frac{a^{3}+a^{2}-2 a}{a|a+2|-a^{2}+4}=\frac{a\left(a^{2}-1\right)+a(a-1)}{a|a+2|-a^{2}+4}=\frac{a(a+2)(a-1)}{a|a+2|-(a-2)(a+2)}
$$
$$
=\left\{\begin{array}{l}
\frac{a(a+2)(a-1)}{-a(a+2)-(a-2)(a+2)}=-\frac{a(a+2)(a-1)}{(a+2)(a+a-2)}=-\frac{a}{2}, \text { if } a<-2 .
\en... | -\frac{}{2}when\in(-\infty,-2);\frac{(-1)}{2}when\in(-2,\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,849 |
2.171. $\frac{\frac{x+y}{\sqrt{x}-\sqrt{y}}-\frac{x-y}{\sqrt{x}+\sqrt{y}}}{\frac{\sqrt{x}-\sqrt{y}}{x+y}+\frac{\sqrt{x}+\sqrt{y}}{x-y}} \cdot \frac{y-\sqrt{x y}+x}{2 \sqrt{x y}}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}x>0, \\ y>0, \\ x \neq y\end{array}\right.$
$$
\begin{aligned}
& \frac{\frac{x+y}{\sqrt{x}-\sqrt{y}}-\frac{x-y}{\sqrt{x}+\sqrt{y}}}{\frac{\sqrt{x}-\sqrt{y}}{x+y}+\frac{\sqrt{x}+\sqrt{y}}{x-y}} \cdot \frac{y-\sqrt{x y}+x}{2 \sqrt{x y}}= \\
& =\frac{\frac{(x+y)(\s... | \frac{x+y}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,850 |
2.174. $\frac{x^{3}-6 x^{2}+11 x+6}{\left(x^{3}-4 x^{2}+3 x\right) \cdot|x-2|}$. | Solution.
ODZ: $\left\{\begin{array}{l}x \neq 0, \\ x \neq 1, \\ x \neq 2, \\ x \neq 3 .\end{array}\right.$
$\frac{x^{3}-6 x^{2}+11 x-6}{\left(x^{3}-4 x^{2}+3 x\right) \cdot|x-2|}=\frac{(x-3)(x-2)(x-1)}{x(x-3)(x-1) \cdot|x-2|}=\frac{x-2}{x \cdot|x-2|}=$
$=\left\{\begin{array}{l}-\frac{x-2}{x(x-2)}=-\frac{1}{x} \text ... | -\frac{1}{x}whenx\in(-\infty;0)\cup(0;1)\cup(1;2);\frac{1}{x}whenx\in(2;3)\cup(3;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,853 |
2.175. $\frac{\sqrt{x-2 \sqrt{x-1}}}{\sqrt{x-1}-1}$. | Solution.
Domain of definition: $1 \leq x \neq 2$.
$$
\begin{aligned}
& \frac{\sqrt{x-2 \sqrt{x-1}}}{\sqrt{x-1}-1}=\frac{\sqrt{x-1-2 \sqrt{x-1}+1}}{\sqrt{x-1}-1}=\frac{\sqrt{(\sqrt{x-1})^{2}-2 \sqrt{x-1}+1}}{\sqrt{x-1}-1}= \\
& =\frac{\sqrt{(\sqrt{x-1}-1)^{2}}}{\sqrt{x-1}-1}=\frac{|\sqrt{x-1}-1|}{\sqrt{x-1}-1}= \\
& ... | -1forx\in[1;2);1forx\in(2;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,854 |
2.176. $\frac{a^{2}-4-|a-2|}{a^{3}+2 a^{2}-5 a-6}$.
2.176. $\frac{a^{2}-4-|a-2|}{a^{3}+2 a^{2}-5 a-6}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}a \neq 2, \\ a \neq-3, \\ a \neq-1 .\end{array}\right.$
$$
\begin{aligned}
& \frac{a^{2}-4-|a-2|}{a^{3}+2 a^{2}-5 a-6}=\frac{(a-2)(a+2)-|a-2|}{(a-2)(a+3)(a+1)}= \\
& =\left\{\begin{aligned}
& \frac{(a-2)(a+2)+(a-2)}{(a-2)(a+3)(a+1)}=\frac{(a-2)(a+2+1)}{(a-2)(a+3... | \frac{1}{+1}for\in(-\infty;-3)\cup(-3;-1)\cup(-1;2);\frac{1}{+3}for\in(2;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,855 |
2.178. $\frac{a^{3}-2 a^{2}+5 a+26}{a^{3}-5 a^{2}+17 a-13}$. | Solution.
Domain of definition: $a \neq 1$.
$$
\begin{aligned}
& \frac{a^{3}-2 a^{2}+5 a+26}{a^{3}-5 a^{2}+17 a-13}=\frac{\left(a^{3}+2 a^{2}\right)-\left(4 a^{2}+8 a\right)+(13 a+26)}{\left(a^{3}-a^{2}\right)-\left(4 a^{2}-8 a\right)+(13 a-13)}= \\
& =\frac{a^{2}(a+2)-4 a(a+2)+13(a+2)}{a^{2}(a-1)-4 a(a-1)+13(a-1)}=\... | \frac{+2}{-1} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,856 |
2.179. $\frac{2 a^{4}+a^{3}+4 a^{2}+a+2}{2 a^{3}-a^{2}+a-2}$. | ## Solution.
Domain of definition: $a \neq 1$.
$$
\begin{aligned}
& \frac{2 a^{4}+a^{3}+4 a^{2}+a+2}{2 a^{3}-a^{2}+a-2}=\frac{\left(2 a^{4}+2 a^{2}\right)+\left(a^{3}+a\right)+\left(2 a^{2}+2\right)}{\left(2 a^{3}-2 a^{2}\right)+\left(a^{2}-a\right)+(2 a-2)}= \\
& =\frac{2 a^{2}\left(a^{2}+1\right)+a\left(a^{2}+1\rig... | \frac{^{2}+1}{-1} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,857 |
2.180. $\frac{|x-1|+|x|+x}{3 x^{2}-4 x+1}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}x \neq \frac{1}{3}, \\ x \neq 1 .\end{array}\right.$
$$
\frac{|x-1|+|x|+x}{3 x^{2}-4 x+1}=\left\{\begin{array}{l}
\frac{-x+1-x+x}{(x-1)(3 x-1)}=-\frac{x-1}{(x-1)(3 x-1)}=\frac{1}{1-3 x} \text { for } x \in(-\infty ; 0) \\
\frac{-x+1+x+x}{(x-1)(3 x-1)}=\frac{x+1}... | \frac{1}{1-3x}forx\in(-\infty;0);\frac{x+1}{(x-1)(3x-1)}forx\in[0;\frac{1}{3})\cup(\frac{1}{3};1);\frac{1}{x-1}forx | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,858 |
2.182. $\frac{\left(a b\left(x^{2}+y^{2}\right)+x y\left(a^{2}+b^{2}\right)\right) \cdot\left((a x+b y)^{2}-4 a b x y\right)}{a b\left(x^{2}-y^{2}\right)+x y\left(a^{2}-b^{2}\right)}$. | Solution.
Domain of definition: $a b\left(x^{2}-y^{2}\right)+x y\left(a^{2}-b^{2}\right) \neq 0, a b x^{2}+\left(a^{2}-b^{2}\right) y x-a b y^{2} \neq 0$,
$$
\begin{aligned}
& x_{1,2} \neq \frac{-\left(a^{2}-b^{2}\right) y \pm y \sqrt{\left(a^{2}-b^{2}\right)^{2} y^{2}+4 a^{2} b^{2} y^{2}}}{2 a b}= \\
& =\frac{-\left... | ^{2}x^{2}-b^{2}y^{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,860 |
2.183. $\frac{x \cdot|x-3|+x^{2}-9}{2 x^{3}-3 x^{2}-9 x}$. | Solution.
Domain of definition: $2 x^{3}-3 x^{2}-9 x \neq 0, x\left(2 x^{2}-3 x-9\right) \neq 0,2 x\left(x+\frac{3}{2}\right)(x-3) \neq 0$,
or $\left\{\begin{array}{l}x \neq 0, \\ x \neq-\frac{3}{2}, \\ x \neq 3 .\end{array}\right.$
$$
\begin{aligned}
& \frac{x \cdot|x-3|+x^{2}-9}{2 x^{3}-3 x^{2}-9 x}=\frac{x \cdot|... | \frac{3}{x(2x+3)},ifx\in(-\infty;-\frac{3}{2})\cup(-\frac{3}{2};0)\cup(0;3);\frac{1}{x},ifx\in(3;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,861 |
2.185. $\frac{x^{2}-1+|x+1|}{|x| \cdot(x-2)}$.
2.185. $\frac{x^{2}-1+|x+1|}{|x| \cdot(x-2)}$. | ## Solution.
odZ: $\left\{\begin{array}{l}x \neq 0, \\ x \neq 2 .\end{array}\right.$
$$
\frac{x^{2}-1+|x+1|}{|x| \cdot(x-2)}=
$$
$$
=\left\{\begin{array}{l}
\frac{x^{2}-1-(x+1)}{-x(x-2)}=\frac{x^{2}-x-2}{-x(x-2)}=\frac{(x-2)(x+1)}{-x(x-2)}=-\frac{x+1}{x} \text { when } x \in(-\infty ;-1) ; \\
\frac{x^{2}-1+(x+1)}{-x... | -\frac{x+1}{x}ifx\in(-\infty,-1);\frac{x+1}{2-x}ifx\in[-1,0);\frac{x+1}{x-2}ifx\in(0,2)\cup(2,\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,862 |
2.186. $\frac{p^{3}+4 p^{2}+10 p+12}{p^{3}-p^{2}+2 p+16} \cdot \frac{p^{3}-3 p^{2}+8 p}{p^{2}+2 p+6}$. | ## Solution.
Domain of definition: $p \neq-2$.
Factor the numerator of the first fraction:
$$
\begin{aligned}
& p^{3}+4 p^{2}+10 p+12=\left(p^{3}+2 p^{2}\right)+\left(2 p^{2}+4 p\right)+(6 p+12)=p^{2}(p+2)+ \\
& +2 p(p+2)+6(p+2)=(p+2)\left(p^{2}+2 p+6\right)
\end{aligned}
$$
Similarly for the denominator, we find t... | p | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,863 |
2.187. $\frac{1+2 a^{1 / 4}-a^{1 / 2}}{1-a+4 a^{3 / 4}-4 a^{1 / 2}}+\frac{a^{1 / 4}-2}{\left(a^{1 / 4}-1\right)^{2}}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}a \neq 1, \\ a \geq 0, \\ a \neq 1 \pm \sqrt{2} .\end{array}\right.$
$$
\frac{1+2 a^{1 / 4}-a^{1 / 2}}{1-a+4 a^{3 / 4}-4 a^{1 / 2}}+\frac{a^{1 / 4}-2}{\left(a^{1 / 4}-1\right)^{2}}=\frac{-\left(a^{2 / 4}-2 a^{1 / 4}-1\right)}{-\left(a^{4 / 4}-1\right)+4\left(a^{... | \frac{1}{\sqrt[4]{}-1} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,864 |
2.190. $\left(\frac{z-2}{6 z+(z-2)^{2}}+\frac{(z+4)^{2}-12}{z^{3}-8}-\frac{1}{z-2}\right): \frac{z^{3}+2 z^{2}+2 z+4}{z^{3}-2 z^{2}+2 z-4}$. | ## Solution.
Domain of definition: $z \neq+2$.
$$
\begin{aligned}
& \left(\frac{z-2}{6 z+(z-2)^{2}}+\frac{(z+4)^{2}-12}{z^{3}-8}-\frac{1}{z-2}\right): \frac{z^{3}+2 z^{2}+2 z+4}{z^{3}-2 z^{2}+2 z-4}= \\
& =\left(\frac{z-2}{6 z+z^{2}-4 z+4}+\frac{z^{2}+8 z+16-12}{(z-2)\left(z^{2}+2 z+4\right)}-\frac{1}{z-2}\right): \f... | \frac{1}{z+2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,867 |
2.191. $\frac{\sqrt{\sqrt{5}-2} \cdot \sqrt[4]{9+4 \sqrt{5}}+\sqrt[3]{a^{2}}-\sqrt[3]{a}}{\sqrt{\sqrt{5}+2} \cdot \sqrt[4]{9-4 \sqrt{5}}+a}$. | Solution.
Domain of definition: $a \neq-1$.
$$
\begin{aligned}
& \frac{\sqrt{\sqrt{5}-2} \cdot \sqrt[4]{9+4 \sqrt{5}}+\sqrt[3]{a^{2}}-\sqrt[3]{a}}{\sqrt{\sqrt{5}+2} \cdot \sqrt[4]{9-4 \sqrt{5}}+a}=\frac{\sqrt{\sqrt{5}-2} \cdot \sqrt[4]{5+4 \sqrt{5}+4}+\sqrt[3]{a^{2}}-\sqrt[3]{a}}{\sqrt{\sqrt{5}+2} \cdot \sqrt[4]{5-4 ... | \frac{1}{1+\sqrt[3]{}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,868 |
2.194. $\frac{\sqrt[3]{\sqrt{5}-\sqrt{3}} \cdot \sqrt[6]{8+2 \sqrt{15}}-\sqrt[3]{a}}{\sqrt[3]{\sqrt{20}+\sqrt{12}} \cdot \sqrt[6]{8-2 \sqrt{15}}-2 \sqrt[3]{2 a}+\sqrt[3]{a^{2}}}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly... | Solution.
Domain of definition: $a \neq 2$.
$$
\begin{aligned}
& \frac{\sqrt[3]{\sqrt{5}-\sqrt{3}} \cdot \sqrt[6]{8+2 \sqrt{15}}-\sqrt[3]{a}}{\sqrt[3]{\sqrt{20}+\sqrt{12}} \cdot \sqrt[6]{8-2 \sqrt{15}}-2 \sqrt[3]{2 a}+\sqrt[3]{a^{2}}}= \\
& =\frac{\sqrt[3]{\sqrt{5}-\sqrt{3}} \cdot \sqrt[6]{5+2 \sqrt{5 \cdot 3}+3}-\sq... | \frac{1}{\sqrt[3]{2}-\sqrt[3]{}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,871 |
2.196. $\frac{\left|x^{2}-1\right|+x^{2}}{2 x^{2}-1}-\frac{|x-1|}{x-1}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}x \neq \pm \frac{\sqrt{2}}{2}, \\ x \neq 1 .\end{array}\right.$
Expanding the absolute values with consideration of the domain of definition, we consider three cases:
1) $\left\{\begin{array}{l}x \in(-\infty ;-1), \\ \frac{x^{2}-1+x^{2}}{2 x^{2}-1}+\frac{x-1}{x... | 2, | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,873 |
2.198. $\frac{b^{2}-3 b-(b-1) \sqrt{b^{2}-4}+2}{b^{2}+3 b-(b+1) \sqrt{b^{2}-4}+2} \cdot \sqrt{\frac{b+2}{b-2}} ; b>2$. | Solution.
$$
\begin{aligned}
& \frac{b^{2}-3 b-(b-1) \sqrt{b^{2}-4}+2}{b^{2}+3 b-(b+1) \sqrt{b^{2}-4}+2} \cdot \sqrt{\frac{b+2}{b-2}}= \\
& =\frac{\left(b^{2}-3 b+2\right)-(b-1) \sqrt{(b-2)(b+2)}}{\left(b^{2}+3 b+2\right)-(b+1) \sqrt{(b-2)(b+2)}} \cdot \sqrt{\frac{b+2}{b-2}}= \\
& =\frac{(b-2)(b-1)-(b-1) \sqrt{(b-2)(b... | \frac{1-b}{1+b} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,875 |
2.199. $\left(\frac{\sqrt[3]{m n^{2}}+\sqrt[3]{m^{2} n}}{\sqrt[3]{m^{2}}+2 \sqrt[3]{m n}+\sqrt[3]{n^{2}}}-2 \sqrt[3]{n}+\frac{m-n}{\sqrt[3]{m^{2}}-\sqrt[3]{n^{2}}}\right):(\sqrt[6]{m}+\sqrt[6]{n})$. | ## Solution.
Domain of definition: $\left\{\begin{array}{l}m \neq n, \\ m \geq 0, \\ n \geq 0 .\end{array}\right.$
$$
\begin{aligned}
& \left(\frac{\sqrt[3]{m n^{2}}+\sqrt[3]{m^{2} n}}{\sqrt[3]{m^{2}}+2 \sqrt[3]{m n}+\sqrt[3]{n^{2}}}-2 \sqrt[3]{n}+\frac{m-n}{\sqrt[3]{m^{2}}-\sqrt[3]{n^{2}}}\right):(\sqrt[6]{m}+\sqrt[... | \sqrt[6]{}-\sqrt[6]{n} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,876 |
2.200. $\left(\frac{\sqrt[4]{x^{3}}-y}{\sqrt[4]{x}-\sqrt[3]{y}}-3 \sqrt[12]{x^{3} y^{4}}\right)^{-1 / 2} \cdot\left(\frac{\sqrt[4]{x^{3}}+y}{\sqrt[4]{x}+\sqrt[3]{y}}-\sqrt[3]{y^{2}}\right)$ | ## Solution.
Domain of definition: $\left\{\begin{array}{l}x \geq 0, \\ \sqrt[3]{y} \neq \pm \sqrt[4]{x} .\end{array}\right.$
$$
\begin{aligned}
& \left(\frac{\sqrt[4]{x^{3}}-y}{\sqrt[4]{x}-\sqrt[3]{y}}-3 \sqrt[12]{x^{3} y^{4}}\right)^{-1 / 2} \cdot\left(\frac{\sqrt[4]{x^{3}}+y}{\sqrt[4]{x}+\sqrt[3]{y}}-\sqrt[3]{y^{2... | \sqrt[4]{x}if\sqrt[4]{x}-\sqrt[3]{y}>0;-\sqrt[4]{x}if\sqrt[4]{x}-\sqrt[3]{y}<0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,877 |
2.201. $\sqrt{\frac{p^{2}-q \sqrt{p}}{\sqrt{p}-\sqrt[3]{q}}+p \sqrt[3]{q}} \cdot\left(p+\sqrt[6]{p^{3} q^{2}}\right)^{-1 / 2}$. | Solution.
Given: $\left\{\begin{array}{l}p>0, \\ \pm \sqrt{p} \neq \sqrt[3]{q}, \\ \sqrt{p}+\sqrt[3]{q}>0 .\end{array}\right.$
$$
\begin{aligned}
& \sqrt{\frac{p^{2}-q \sqrt{p}}{\sqrt{p}-\sqrt[3]{q}}+p \sqrt[3]{q}} \cdot\left(p+\sqrt[6]{p^{3} q^{2}}\right)^{-1 / 2}= \\
& =\sqrt{\frac{\sqrt{p}\left(\sqrt{p^{3}}-q\righ... | \sqrt{\sqrt{p}+\sqrt[3]{q}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,878 |
2.202. $\frac{\sqrt[3]{m+4 \sqrt{m-4}} \cdot \sqrt[3]{\sqrt{m-4}+2}}{\sqrt[3]{m-4 \sqrt{m-4}} \cdot \sqrt[3]{\sqrt{m-4}-2}} \cdot \frac{m-4 \sqrt{m-4}}{2}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}m \geq 4, \\ m \neq 8 .\end{array}\right.$
$$
\begin{aligned}
& \sqrt[3]{m+4 \sqrt{m-4}} \cdot \sqrt[3]{\sqrt{m-4}+2} \cdot \frac{m-4 \sqrt{m-4}}{2}= \\
& =\frac{\sqrt[3]{m-4+4 \sqrt{m-4}+4}}{\sqrt[3]{m-4-4 \sqrt{m-4}+4} \cdot \sqrt[3]{\sqrt{m-4}+2}} \cdot \frac... | \frac{-8}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,879 |
2.203. $\frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2} \cdot\left(2 x+\sqrt{x^{2}-1}\right)}{\sqrt{(x+1)^{3}}-\sqrt{(x-1)^{3}}}$.
2.203. $\frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2} \cdot\left(2 x+\sqrt{x^{2}-1}\right)}{\sqrt{(x+1)^{3}}-\sqrt{(x-1)^{3}}}$.
The expression is already in a... | ## Solution.
Domain of definition: $x>1$.
$$
\begin{aligned}
& \frac{\sqrt{\sqrt{\frac{x-1}{x+1}}+\sqrt{\frac{x+1}{x-1}}-2} \cdot\left(2 x+\sqrt{x^{2}-1}\right)}{\sqrt{(x+1)^{3}}-\sqrt{(x-1)^{3}}}= \\
& =\frac{\sqrt{\frac{\sqrt{x-1}}{\sqrt{x+1}}+\frac{\sqrt{x+1}}{\sqrt{x-1}}-2} \cdot\left(x+1+\sqrt{x^{2}-1}+x-1\right... | \frac{1}{\sqrt[4]{x^{2}-1}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,880 |
2.204. $\sqrt{2+\sqrt{3}} \cdot \sqrt{2+\sqrt{2+\sqrt{3}}} \cdot \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}} \cdot \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}$. | Solution.
$$
\begin{aligned}
& \sqrt{2+\sqrt{3}} \cdot \sqrt{2+\sqrt{2+\sqrt{3}}} \cdot \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}} \cdot \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}= \\
& =\sqrt{2+\sqrt{3}} \cdot \sqrt{2+\sqrt{2+\sqrt{3}}} \cdot \sqrt{(2+\sqrt{2+\sqrt{2+\sqrt{3}}})(2-\sqrt{2+\sqrt{2+\sqrt{3}}})}= \\
& =\sqrt{2+\sqrt{... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,881 |
2.205. $\left(\frac{b x+4+\frac{4}{b x}}{2 b+\left(b^{2}-4\right) x-2 b x^{2}}+\frac{\left(4 x^{2}-b^{2}\right) \cdot \frac{1}{b}}{(b+2 x)^{2}-8 b x}\right) \cdot \frac{b x}{2}$. | Solution.
odZ: $\left\{\begin{array}{l}b \neq 0, \\ x \neq 0, \\ x \neq \frac{b}{2}, \\ x \neq-\frac{2}{b} .\end{array}\right.$
$$
\begin{aligned}
& \left(\frac{b x+4+\frac{4}{b x}}{2 b+\left(b^{2}-4\right) x-2 b x^{2}}+\frac{\left(4 x^{2}-b^{2}\right) \cdot \frac{1}{b}}{(b+2 x)^{2}-8 b x}\right) \cdot \frac{b x}{2}=... | \frac{x^{2}-1}{2x-b} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,882 |
2.208. $\frac{\left((\sqrt[4]{m}+\sqrt[4]{n})^{2}-(\sqrt[4]{m}-\sqrt[4]{n})^{2}\right)^{2}-(16 m+4 n)}{4 m-n}+\frac{10 \sqrt{m}-3 \sqrt{n}}{\sqrt{n}+2 \sqrt{m}}$. | ## Solution.
Domain of definition: $\left\{\begin{array}{l}n \neq 4 m, \\ m>0, \\ n>0 .\end{array}\right.$
$$
\frac{\left((\sqrt[4]{m}+\sqrt[4]{n})^{2}-(\sqrt[4]{m}-\sqrt[4]{n})^{2}\right)^{2}-(16 m+4 n)}{4 m-n}+\frac{10 \sqrt{m}-3 \sqrt{n}}{\sqrt{n}+2 \sqrt{m}}=
$$
$=\frac{(\sqrt{m}+2 \sqrt[4]{m n}+\sqrt{n}-\sqrt{m... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,885 |
2.209. $\left(\frac{x-9}{x+3 x^{1 / 2}+9}: \frac{x^{0.5}+3}{x^{1.5}-27}\right)^{0.5}-x^{0.5}$. | ## Solution.
Domain of definition: $0 \leq x \neq 9$.
$$
\begin{aligned}
& \left(\frac{x-9}{x+3 x^{1 / 2}+9}: \frac{x^{0.5}+3}{x^{1.5}-27}\right)^{0.5}-x^{0.5}=\left(\frac{x-9}{(\sqrt{x})^{2}+3 \sqrt{x}+9}: \frac{\sqrt{x}+3}{(\sqrt{x})^{3}-3^{3}}\right)^{0.5}- \\
& -\sqrt{x}=\sqrt{\frac{(\sqrt{x})^{2}-3^{2}}{(\sqrt{x... | 3-2\sqrt{x}ifx\in[0;9);-3ifx\in(9;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,886 |
2.213. $\frac{\sqrt{1+\left(\frac{x^{2}-1}{2 x}\right)^{2}}}{\left(x^{2}+1\right) \cdot \frac{1}{x}}$. | Solution.
Domain of definition: $\boldsymbol{x} \neq 0$.
$$
\begin{aligned}
& \frac{\sqrt{1+\left(\frac{x^{2}-1}{2 x}\right)^{2}}}{\left(x^{2}+1\right) \cdot \frac{1}{x}}=\frac{\sqrt{1+\frac{x^{4}-2 x^{2}+1}{4 x^{2}}}}{\frac{x^{2}+1}{x}}=\frac{\sqrt{\frac{4 x^{2}+x^{4}-2 x^{2}+1}{4 x^{2}}}}{\frac{x^{2}+1}{x}}=
\end{a... | -\frac{1}{2},ifx\in(-\infty;0);\frac{1}{2},ifx\in(0;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,890 |
2.214. $\frac{x^{2}+4}{x \sqrt{4+\left(\frac{x^{2}-4}{2 x}\right)^{2}}}$.
2.214. $\frac{x^{2}+4}{x \sqrt{4+\left(\frac{x^{2}-4}{2 x}\right)^{2}}}$.
(Note: The mathematical expression is the same in both languages, so the translation is identical to the original text.) | Solution.
Domain of definition: $x \neq 0$.
$$
\begin{aligned}
& \frac{x^{2}+4}{x \sqrt{4+\left(\frac{x^{2}-4}{2 x}\right)^{2}}}=\frac{x^{2}+4}{x \sqrt{4+\frac{x^{4}-8 x^{2}+16}{4 x^{2}}}}=\frac{x^{2}+4}{x \sqrt{\frac{16 x^{2}+x^{4}-8 x^{2}+16}{4 x^{2}}}}= \\
& =\frac{x^{2}+4}{x \sqrt{\frac{x^{4}+8 x^{2}+16}{4 x^{2}}... | -2,ifx\in(-\infty;0);2,ifx\in(0;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,891 |
2.215. $\left((z-3)(z+3)^{-1}-\frac{(z+3)^{3 / 2}}{\sqrt{\left(z^{2}-9\right)(z-3)}}\right) \cdot \frac{\frac{1}{3}-\frac{z}{18}-\frac{1}{2 z}}{(z+3)^{-1}}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}z \neq 0, \\ z>-3, \\ z \neq 3\end{array}\right.$
$$
\begin{aligned}
& \left((z-3)(z+3)^{-1}-\frac{(z+3)^{3 / 2}}{\sqrt{\left(z^{2}-9\right)(z-3)}}\right) \cdot \frac{\frac{1}{3}-\frac{z}{18}-\frac{1}{2 z}}{(z+3)^{-1}}= \\
& =\left(\frac{z-3}{z+3}-\frac{\sqrt{(z... | \frac{(z^{2}+9)(3-z)}{9z},ifz\in(-3;0)\cup(0;3);\frac{2(z-3)}{3},ifz\in(3;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,892 |
2.216. $\frac{\sqrt{\frac{m+2}{m-2}}+\sqrt{\frac{m-2}{m+2}}}{\sqrt{\frac{m+2}{m-2}}-\sqrt{\frac{m-2}{m+2}}}$.
2.216. $\frac{\sqrt{\frac{m+2}{m-2}}+\sqrt{\frac{m-2}{m+2}}}{\sqrt{\frac{m+2}{m-2}}-\sqrt{\frac{m-2}{m+2}}}$.
Note: The mathematical expression is the same in both languages, so the translation is identical. | Solution.
Domain of definition: $\left\{\begin{array}{l}m>2, \\ m<-2 .\end{array}\right.$
$$
\begin{aligned}
& \frac{\sqrt{\frac{m+2}{m-2}}+\sqrt{\frac{m-2}{m+2}}}{\sqrt{\frac{m+2}{m-2}}-\sqrt{\frac{m-2}{m+2}}}=\frac{\left(\sqrt{\frac{m+2}{m-2}}+\sqrt{\frac{m-2}{m+2}}\right)\left(\sqrt{\frac{m+2}{m-2}}+\sqrt{\frac{m-... | \frac{}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,893 |
2.217. $\frac{b^{-1 / 6} \cdot \sqrt{a^{3} b} \cdot \sqrt[3]{a^{3} b}-\sqrt{a^{3} b^{2}} \cdot \sqrt[3]{b^{2}}}{\left(2 a^{2}-b^{2}-a b\right) \cdot \sqrt[6]{a^{9} b^{4}}}:\left(\frac{3 a^{3}}{2 a^{2}-a b-b^{2}}-\frac{a b}{a-b}\right)$ | ## Solution.
Domain of definition: $\left\{\begin{array}{l}b \neq 0, \\ a b>0, \\ a \neq b, \\ a \neq-\frac{b}{.2} .\end{array}\right.$
$$
\begin{aligned}
& \frac{b^{-1 / 6} \cdot \sqrt{a^{3} b} \cdot \sqrt[3]{a^{3} b}-\sqrt{a^{3} b^{2}} \cdot \sqrt[3]{b^{2}}}{\left(2 a^{2}-b^{2}-a b\right) \cdot \sqrt[6]{a^{9} b^{4}... | \frac{1}{(3+b)} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,894 |
2.219. $\left(\frac{9}{a+8}-\frac{a^{1 / 3}+2}{a^{2 / 3}-2 a^{1 / 3}+4}\right) \cdot \frac{a^{4 / 3}+8 a^{1 / 3}}{1-a^{2 / 3}}+\frac{5-a^{2 / 3}}{1+a^{1 / 3}}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}a \neq-8, \\ a \neq \pm 1 .\end{array}\right.$
$$
\begin{aligned}
& \left(\frac{9}{a+8}-\frac{a^{1 / 3}+2}{a^{2 / 3}-2 a^{1 / 3}+4}\right) \cdot \frac{a^{4 / 3}+8 a^{1 / 3}}{1-a^{2 / 3}}+\frac{5-a^{2 / 3}}{1+a^{1 / 3}}= \\
& =\left(\frac{9}{\left(a^{1 / 3}+2\rig... | 5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,895 |
2.221. $\frac{\left.\sqrt{1+\sqrt{1-x^{2}}} \cdot(\sqrt{(1+x})^{3}-\sqrt{(1-x)^{3}}\right)}{2+\sqrt{1-x^{2}}}$.
2.221. $\frac{\left.\sqrt{1+\sqrt{1-x^{2}}} \cdot(\sqrt{(1+x})^{3}-\sqrt{(1-x)^{3}}\right)}{2+\sqrt{1-x^{2}}}$.
(Note: The original text and the translation are identical as the expression is already in a u... | ## Solution.
Domain of definition: $-1 \leq x \leq 1$.
$$
\begin{aligned}
& \frac{\sqrt{1+\sqrt{1-x^{2}}} \cdot\left(\sqrt{(1+x)^{3}}-\sqrt{(1-x)^{3}}\right)}{2+\sqrt{1-x^{2}}}= \\
& =\frac{\sqrt{1+\sqrt{1-x^{2}}} \cdot(\sqrt{1+x}-\sqrt{1-x})(1+x+\sqrt{(1+x)(1-x)}+1-x)}{2+\sqrt{1-x^{2}}}= \\
& =\frac{\sqrt{1+\sqrt{1-... | x\sqrt{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,897 |
2.222. $\left(\frac{2-n}{n-1}+4 \cdot \frac{m-1}{m-2}\right):\left(n^{2} \cdot \frac{m-1}{n-1}+m^{2} \cdot \frac{2-n}{m-2}\right) ; m=\sqrt[4]{400}, n=\sqrt{5}$. | Solution.
$$
\begin{aligned}
& \left(\frac{2-n}{n-1}+4 \cdot \frac{m-1}{m-2}\right):\left(n^{2} \cdot \frac{m-1}{n-1}+m^{2} \cdot \frac{2-n}{m-2}\right)= \\
& =\frac{(2-n)(m-2)+4(m-1)(n-1)}{(n-1)(m-2)}: \frac{n^{2}(m-1)(m-2)+m^{2}(2-n)(n-1)}{(n-1)(m-2)}= \\
& =\frac{3 m n-2(m+n)}{(n-1)(m-2)}: \frac{(m-n)(3 m n-2(m+n))... | \frac{\sqrt{5}}{5} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,898 |
2.223. $\frac{\sqrt{\frac{1}{a+2 \sqrt{a-2}-1}}+\sqrt{\frac{1}{a-2 \sqrt{a-2}-1}}}{\sqrt{\frac{1}{a+2 \sqrt{a-2}-1}}-\sqrt{\frac{1}{a-2 \sqrt{a-2}-1}}}$.
2.223. $\frac{\sqrt{\frac{1}{a+2 \sqrt{a-2}-1}}+\sqrt{\frac{1}{a-2 \sqrt{a-2}-1}}}{\sqrt{\frac{1}{a+2 \sqrt{a-2}-1}}-\sqrt{\frac{1}{a-2 \sqrt{a-2}-1}}}$.
The expres... | Solution.
odZ: $\left\{\begin{array}{l}a>2, \\ a \neq 3 .\end{array}\right.$
$\frac{\sqrt{\frac{1}{a+2 \sqrt{a-2}-1}}+\sqrt{\frac{1}{a-2 \sqrt{a-2}-1}}}{\sqrt{\frac{1}{a+2 \sqrt{a-2}-1}}-... | -\frac{1}{\sqrt{-2}},if\in(2;3);-\sqrt{-2},if\in(3;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,899 |
2.224. $\frac{1}{\sqrt{x^{2}+4 x+4}}+|x-2|$. | ## Solution.
Domain of definition: $x \neq-2$.
$$
\begin{aligned}
& \frac{1}{\sqrt{x^{2}+4 x+4}}+|x-2|=\frac{1}{\sqrt{(x+2)^{2}}}+|x-2|=\frac{1}{|x+2|}+|x-2|= \\
& =\frac{1+|x+2| \cdot|x-2|}{|x+2|}=\frac{1+|(x+2)(x-2)|}{|x+2|}=\frac{1+\left|x^{2}-4\right|}{|x+2|}= \\
& =\left\{\begin{array}{l}
\frac{1+x^{2}-4}{-(x+2)... | \frac{3-x^{2}}{x+2}ifx\in(-\infty;-2);\frac{5-x^{2}}{x+2}ifx\in(-2;2);\frac{x^{2}-3}{x+2}ifx\in[2;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,900 |
2.225. $\left(x^{2}-6 x+1+\left(\frac{\frac{x-3}{1+3 x}-\frac{x-5}{1+5 x}}{1+\frac{(x-5)(x-3)}{(1+5 x)(1+3 x)}}\right)^{-1}\right)^{1 / 2}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}x \neq-\frac{1}{3}, \\ x \neq-\frac{1}{5} .\end{array}\right.$
$$
\begin{aligned}
& \left(x^{2}-6 x+1+\left(\frac{\frac{x-3}{1+3 x}-\frac{x-5}{1+5 x}}{1+\frac{(x-5)(x-3)}{(1+5 x)(1+3 x)}}\right)^{-1}\right)^{1 / 2}= \\
& =\left(x^{2}-6 x+1+\left(\frac{\frac{(x-3... | |x-3| | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,901 |
2.226. $\left(\frac{1}{(x+3)^{2}} \cdot\left(\frac{1}{x^{2}}+\frac{1}{9}\right)+\frac{2}{(x+3)^{3}} \cdot\left(\frac{1}{x}+\frac{1}{3}\right)\right)^{-1 / 2} \cdot$ | ## Solution.
Domain of definition: $\left\{\begin{array}{l}x \neq 0, \\ x \neq-3 .\end{array}\right.$
$$
\begin{aligned}
& \left(\frac{1}{(x+3)^{2}} \cdot\left(\frac{1}{x^{2}}+\frac{1}{9}\right)+\frac{2}{(x+3)^{3}} \cdot\left(\frac{1}{x}+\frac{1}{3}\right)\right)^{-1 / 2}= \\
& =\left(\frac{1}{(x+3)^{2}} \cdot \frac{... | -3x,ifx\in(-\infty;-3)\cup(-3;0);3x,ifx\in(0;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,902 |
2.227. $\frac{\sqrt{2 a+2 \sqrt{a^{2}-9}}}{\sqrt{2 a-2 \sqrt{a^{2}-9}}}$.
2.227. $\frac{\sqrt{2 a+2 \sqrt{a^{2}-9}}}{\sqrt{2 a-2 \sqrt{a^{2}-9}}}$. | ## Solution.
Domain of definition: $a \geq 3$.
$\frac{\sqrt{2 a+2 \sqrt{a^{2}-9}}}{\sqrt{2 a-2 \sqrt{a^{2}-9}}}=\frac{\sqrt{a+3+2 \sqrt{(a+3)(a-3)}+a-3}}{\sqrt{a+3-2 \sqrt{(a+3)(a-3)}+a-3}}=$
$=\frac{\sqrt{(\sqrt{a+3}+\sqrt{a-3})^{2}}}{\sqrt{(\sqrt{a+3}-\sqrt{a-3})^{2}}}=\frac{\sqrt{a+3}+\sqrt{a-3}}{\sqrt{a+3}-\sqrt... | \frac{+\sqrt{^{2}-9}}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,903 |
2.228. $\sqrt{\left(y^{2}+\frac{4}{y^{2}}\right)^{2}-8 \cdot\left(y+\frac{2}{y}\right)^{2}+48}$. | Solution.
Domain of definition: $y \neq 0$.
\[
\begin{aligned}
& \sqrt{\left(y^{2}+\frac{4}{y^{2}}\right)^{2}-8\left(y+\frac{2}{y}\right)^{2}+48}=\sqrt{\left(\left(y+\frac{2}{y}\right)^{2}-4\right)^{2}-8\left(y+\frac{2}{y}\right)^{2}+48}= \\
& =\sqrt{\left(y+\frac{2}{y}\right)^{4}-8\left(y+\frac{2}{y}\right)^{2}+16-8... | (y-\frac{2}{y})^{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,904 |
2.229. $\frac{x+\sqrt{3}}{\sqrt{x}+\sqrt{x+\sqrt{3}}}+\frac{x-\sqrt{3}}{\sqrt{x}-\sqrt{x-\sqrt{3}}} ; x=2$. | Solution.
$$
\begin{aligned}
& \frac{x+\sqrt{3}}{\sqrt{x}+\sqrt{x+\sqrt{3}}}+\frac{x-\sqrt{3}}{\sqrt{x}-\sqrt{x-\sqrt{3}}}=\frac{(x+\sqrt{3})(\sqrt{x}-\sqrt{x+\sqrt{3}})}{(\sqrt{x}+\sqrt{x+\sqrt{3}})(\sqrt{x}-\sqrt{x+\sqrt{3}})^{+}}+ \\
& +\frac{(x-\sqrt{3})(\sqrt{x}+\sqrt{x-\sqrt{3}})}{(\sqrt{x}-\sqrt{x-\sqrt{3}})(\s... | \sqrt{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,905 |
2.230. $\frac{\sqrt{x-2 \sqrt{2}}}{\sqrt{x^{2}-4 x \sqrt{2}+8}}-\frac{\sqrt{x+2 \sqrt{2}}}{\sqrt{x^{2}+4 x \sqrt{2}+8}} ; x=3$. | ## Solution.
$$
\begin{aligned}
& \frac{\sqrt{x-2 \sqrt{2}}}{\sqrt{x^{2}-4 x \sqrt{2}+8}}-\frac{\sqrt{x+2 \sqrt{2}}}{\sqrt{x^{2}+4 x \sqrt{2}+8}}=\frac{\sqrt{x-2 \sqrt{2}}}{\sqrt{(x-2 \sqrt{2})^{2}}}-\frac{\sqrt{x+2 \sqrt{2}}}{\sqrt{(x+2 \sqrt{2})^{2}}}= \\
& =\frac{1}{\sqrt{x-2 \sqrt{2}}}-\frac{1}{\sqrt{x+2 \sqrt{2}}... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,906 |
2.231. $\frac{1+z}{1+\sqrt{1+z}}-\frac{1-z}{1-\sqrt{1-z}} ; z=\frac{\sqrt{3}}{2}$. | ## Solution.
$$
\begin{aligned}
& \frac{1+z}{1+\sqrt{1+z}}-\frac{1-z}{1-\sqrt{1-z}}=\frac{(1+z)(1-\sqrt{1+z})}{(1+\sqrt{1+z})(1-\sqrt{1+z})}-\frac{(1-z)(1+\sqrt{1-z})}{(1-\sqrt{1-z})(1+\sqrt{1-z})}= \\
& =\frac{1-\sqrt{1+z}+z-z \sqrt{1+z}}{1-1-z}-\frac{1+\sqrt{1-z}-z-z \sqrt{1-z}}{1-1+z}= \\
& =\frac{1+z-\sqrt{1+z}(1+... | \frac{\sqrt{3}}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,907 |
2.232. $\frac{a^{2}-3}{\sqrt{\left(\frac{a^{2}+3}{2 a}\right)^{2}-3}}$.
2.232. $\frac{a^{2}-3}{\sqrt{\left(\frac{a^{2}+3}{2 a}\right)^{2}-3}}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}a \neq 0, \\ a \neq \pm \sqrt{3} .\end{array}\right.$
$$
\frac{a^{2}-3}{\sqrt{\left(\frac{a^{2}+3}{2 a}\right)^{2}-3}}=\frac{a^{2}-3}{\sqrt{\frac{a^{4}+6 a^{2}+9}{4 a^{2}}-3}}=\frac{a^{2}-3}{\sqrt{\frac{a^{4}+6 a^{2}+9-12 a^{2}}{4 a^{2}}}}=
$$
$$
=\frac{a^{2}-3... | -2,if\in(-\infty;-\sqrt{3})\cup(0;\sqrt{3});2,if\in(-\sqrt{3};0)\cup(\sqrt{3};\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,908 |
2.234. $\frac{1+\sqrt{1+x}}{x+1}+\frac{1+\sqrt{1-x}}{x-1} ; x=\frac{\sqrt{3}}{2}$. | Solution.
$$
\begin{aligned}
& \frac{1+\sqrt{1+x}}{x+1}+\frac{1+\sqrt{1-x}}{x-1}=\frac{x+x \sqrt{1+x}-1-\sqrt{1+x}+x+x \sqrt{1-x}+1+\sqrt{1-x}}{x^{2}-1} \\
& =\frac{2 x-(1-x) \sqrt{1+x}+(1+x) \sqrt{1-x}}{x^{2}-1}= \\
& =\frac{\sqrt{3}-\left(1-\frac{\sqrt{3}}{2}\right) \sqrt{1+\frac{\sqrt{3}}{2}}+\left(1+\frac{\sqrt{3}... | -2-4\sqrt{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,910 |
2.236. $\frac{\sqrt{z^{2}-1}}{\sqrt{z^{2}-1}-z} ; \quad z=\frac{1}{2}\left(\sqrt{m}+\frac{1}{\sqrt{m}}\right)$ | Solution.
Domain of definition: $m>0$.
$$
\begin{aligned}
& \frac{\sqrt{z^{2}-1}}{\sqrt{z^{2}-1}-z}=\frac{\sqrt{z^{2}-1} \cdot\left(\sqrt{z^{2}-1}+z\right)}{\left(\sqrt{z^{2}-1}-z\right)\left(\sqrt{z^{2}-1}+z\right)}=\frac{\sqrt{z^{2}-1} \cdot\left(\sqrt{z^{2}-1}+z\right)}{\left(\sqrt{z^{2}-1}\right)^{2}-z^{2}}= \\
&... | \frac{-1}{2},if\in(0;1);\frac{1-}{2},if\in[1;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,911 |
2.237. $\left(\sqrt[3]{\frac{x+1}{x-1}}+\sqrt[3]{\frac{x-1}{x+1}}-2\right)^{1 / 2} ; x=\frac{a^{3}+1}{a^{3}-1}$. | Solution.
DZ: $\left\{\begin{array}{l}a \neq 1, \\ a>0 .\end{array}\right.$
$$
\left(\sqrt[3]{\frac{x+1}{x-1}}+\sqrt[3]{\frac{x-1}{x+1}}-2\right)^{1 / 2}=\sqrt{\sqrt[3]{\frac{x+1}{x-1}}+\frac{1}{\sqrt[3]{\frac{x+1}{x-1}}}-2}=
$$
$$
\begin{aligned}
& =\sqrt{\frac{\left(\sqrt[3]{\frac{x+1}{x-1}}\right)^{2}-2 \sqrt[3]{... | {\begin{pmatrix}\frac{1-}{\sqrt{}},&} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,912 |
2.239. $\left(\frac{\sqrt[4]{8}+2}{\sqrt[4]{2}+\sqrt[3]{2}}-\sqrt[3]{4}\right):\left(\frac{\sqrt[4]{8}-2}{\sqrt[4]{2}-\sqrt[3]{2}}-3^{12} \sqrt{128}\right)^{1 / 2}$. | ## Solution.
$$
\begin{aligned}
& x=\frac{\sqrt[4]{8}+2}{\sqrt[4]{2}+\sqrt[3]{2}}-\sqrt[3]{4}=\frac{\sqrt[12]{2^{9}}+\sqrt[12]{2^{12}}}{\sqrt[12]{2^{3}}+\sqrt[12]{2^{4}}}-\sqrt[12]{2^{8}}=\frac{\sqrt[12]{\left(2^{3}\right)^{3}}+\sqrt[12]{\left(2^{4}\right)^{3}}}{\sqrt[12]{2^{3}}+\sqrt[12]{2^{4}}}-\sqrt[12]{2^{8}}= \\
... | -\sqrt[4]{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,913 |
2.240. $\frac{\sqrt{\left(\frac{9-2 \sqrt{3}}{\sqrt{3}-\sqrt[3]{2}}+3 \sqrt[3]{2}\right) \cdot \sqrt{3}}}{3+\sqrt[6]{108}}$. | Solution.
$\frac{\sqrt{\left(\frac{9-2 \sqrt{3}}{\sqrt{3}-\sqrt[3]{2}}+3 \sqrt[3]{2}\right) \cdot \sqrt{3}}}{3+\sqrt[6]{108}}=\frac{\sqrt{\left(\frac{3^{2}-\sqrt{2^{2} \cdot 3}}{\sqrt{3}-\sqrt[3]{2}}+\sqrt[3]{3^{3} \cdot 2}\right) \cdot \sqrt{3}}}{3+\sqrt[6]{27 \cdot 4}}=$
$=\frac{\sqrt{\left(\frac{\sqrt[6]{3^{12}}-\... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,914 |
2.241. $\left(\frac{4-2 x+x^{2}}{4-2 x}+\frac{6 x^{2}+8+12 x}{4-x^{2}}-\frac{x^{2}+2 x+4}{2 x+4}\right)^{-1 / 3} \cdot(x+2)$.
2.241. $\left(\frac{4-2x+x^2}{4-2x}+\frac{6x^2+8+12x}{4-x^2}-\frac{x^2+2x+4}{2x+4}\right)^{-1/3} \cdot (x+2)$. | ## Solution.
Domain of definition: $x \neq \pm 2$.
$$
\begin{aligned}
& \left(\frac{4-2 x+x^{2}}{4-2 x}+\frac{6 x^{2}+8+12 x}{4-x^{2}}-\frac{x^{2}+2 x+4}{2 x+4}\right)^{-1 / 3} \cdot(x+2)= \\
& =\left(-\frac{x^{2}-2 x+4}{2(x-2)}-\frac{6 x^{2}+12 x+8}{(x-2)(x+2)}-\frac{x^{2}+2 x+4}{2(x+2)}\right)^{-1 / 3} \cdot(x+2)= ... | \sqrt[3]{4-x^{2}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,915 |
2.242. $\left(\frac{\sqrt{(z+2)^{2}-8 z}}{z+2}+\frac{(z-1)^{2}+3}{z^{3}+8}\right): \frac{z^{2}-3 z+2}{z^{3}-2 z^{2}-4 z+8}$. | Solution.
Domain of definition: $\left\{\begin{array}{l}z \neq \pm 2 \\ z \neq 1\end{array}\right.$
$\left(\frac{\sqrt{(z+2)^{2}-8 z}}{z+2}+\frac{(z-1)^{2}+3}{z^{3}+8}\right): \frac{z^{2}-3 z+2}{z^{3}-2 z^{2}-4 z+8}=$
$=\left(\frac{\sqrt{z^{2}+4 z+4-8 z}}{z+2}+\frac{z^{2}-2 z+1+3}{(z+2)\left(z^{2}-2 z+4\right)}\right... | \frac{z^{2}-5z+6}{1-z},ifz\in(-\infty;-2)\cup(-2;1)\cup(1;2);z-2,ifz\in(2;\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,916 |
2.243. $\left(\frac{x^{4}+5 x^{3}+15 x-9}{x^{6}+3 x^{4}}+\frac{9}{x^{4}}\right): \frac{x^{3}-4 x+3 x^{2}-12}{x^{5}}$. | ## Solution.
Domain of definition: $\left\{\begin{array}{l}x \neq \pm 2, \\ x \neq 0, \\ x \neq-3 .\end{array}\right.$
$\left(\frac{x^{4}+5 x^{3}+15 x-9}{x^{6}+3 x^{4}}+\frac{9}{x^{4}}\right): \frac{x^{3}-4 x+3 x^{2}-12}{x^{5}}=$
$=\left(\frac{x^{4}+5 x^{3}+15 x-9}{x^{4}\left(x^{2}+3\right)}+\frac{9}{x^{4}}\right): ... | \frac{x}{x-2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 47,917 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.