question stringlengths 19 2.25k | detailed_answer stringlengths 176 4.65k | answer stringlengths 1 99 | question_id stringlengths 5 5 | __original_row_idx int64 0 7.46k |
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The fraction \( \frac{c^{-4}-d^{-4}}{c^{-2}-d^{-2}} \) is equal to: \( \textbf{(A)}\ c^{-6}-d^{-6} \quad \textbf{(B)}\ c^{-2}-d^{-2} \quad \textbf{(C)}\ c^{-2}+d^{-2} \quad \textbf{(D)}\ c^2+d^2 \quad \textbf{(E)}\ c^2-d^2 \) | 1. **Rewrite the numerator using the difference of squares formula**: \( c^{-4} - d^{-4} = (c^{-2} - d^{-2})(c^{-2} + d^{-2}) \).\n\n2. **Substitute the rewritten numerator into the fraction**: \( \frac{c^{-4} - d^{-4}}{c^{-2} - d^{-2}} = \frac{(c^{-2} - d^{-2})(c^{-2} + d^{-2})}{c^{-2} - d^{-2}} \).\n\n3. **Simplify... | c^{-2} + d^{-2 | 29873 | 525 |
Calculate \( \sqrt{12} + \sqrt{32} = \) | 1. Simplify \( \sqrt{12} \):\n\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \]\n\n2. Simplify \( \sqrt{32} \):\n\[ \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} \]\n\n3. Add the simplified terms together (note they have different radicals):\n\[ 2\sqrt{3} + 4\sqrt{2} \]\n... | 2\sqrt{3} + 4\sqrt{2} | 29874 | 191 |
Evaluate \( \sqrt{50} - \sqrt{2} = \) | 1. First, simplify \( \sqrt{50} \):\n\[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \]\n\n2. Now substitute back into the expression:\n\[ 5\sqrt{2} - \sqrt{2} \]\n\n3. Combine like terms:\n\[ (5 - 1)\sqrt{2} = 4\sqrt{2} \] | 4\sqrt{2} | 29875 | 191 |
Solve for \( \sqrt{18} + 2\sqrt{2} = \) | 1. Simplify \( \sqrt{18} \):\n\[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \]\n\n2. Substitute back into the equation: \( 3\sqrt{2} + 2\sqrt{2} \)\n\n3. Combine the like terms:\n\[ (3 + 2)\sqrt{2} = 5\sqrt{2} \] | 5\sqrt{2} | 29876 | 191 |
What is the value of $\frac{(5000 - 4875)^2}{25}$? | 1. **Calculate the difference in the numerator**:
$5000 - 4875 = 125$.
2. **Square the difference**:
$(5000 - 4875)^2 = 125^2 = 15625$.
3. **Divide by the denominator**:
$\frac{15625}{25} = 625$.
4. **Conclude with the correct answer**:
$\boxed{625}$. | 625 | 29877 | 426 |
What is the value of $\frac{(3200 - 3150)^2}{16}$? | 1. **Calculate the difference in the numerator**:
$3200 - 3150 = 50$.
2. **Square the difference**:
$(3200 - 3150)^2 = 50^2 = 2500$.
3. **Divide by the denominator**:
$\frac{2500}{16} = 156.25$.
4. **Conclude with the correct answer**:
$\boxed{156.25}$. | 156.25 | 29878 | 426 |
What is the value of $\frac{(720 - 680)^2}{36}$? | 1. **Calculate the difference in the numerator**:
$720 - 680 = 40$.
2. **Square the difference**:
$(720 - 680)^2 = 40^2 = 1600$.
3. **Divide by the denominator**:
$\frac{1600}{36} = 44.44\ldots$.
4. **Conclude with the correct answer**:
$\boxed{44.44}$. | 44.44 | 29879 | 426 |
A prism has a triangular base with vertices labeled from 1 to 6. Each vertex must be assigned a number from 1 to 6 such that the sum of the vertices on each triangular face is the same. How many unique arrangements are possible, considering arrangements that can be obtained from rotations of the prism are the same? | To solve this problem, let's first calculate the total sum of the integers from 1 to 6: 1 + 2 + 3 + 4 + 5 + 6 = 21. Since there are two triangular faces, and each vertex is part of 2 faces, the sum for each face must equal \( \frac{21}{2} = 10.5 \). However, this is incorrect as we require an integer sum. The arrangeme... | 0 | 29880 | 360 |
You have a tetrahedron with vertices labeled from 1 to 4. Each vertex must be labeled with numbers 1 through 4, ensuring the sum of the vertices on any face is the same. What is the number of distinct arrangements considering rotational symmetry? | The sum of the integers from 1 to 4 is 1 + 2 + 3 + 4 = 10. Since there are 4 triangular faces and each vertex contributes to 3 faces, we find that each face must have a sum of \( \frac{10 \times 3}{4} = 7.5 \), which is impossible to achieve with integer labels. Hence, there are no possible arrangements, leading us to ... | 0 | 29881 | 360 |
A square pyramid has a base defined by vertices numbered 1 to 4 and an apex vertex labeled 5. Each labeled vertex must be arranged such that the sum of the vertices on the sides of the pyramid is equal. How many distinct arrangements are possible, counting rotations as the same? | The sum of the labels from 1 to 5 is 1 + 2 + 3 + 4 + 5 = 15. Each of the triangular side faces shares an edge with the base. The average sum for a side must equal \( \frac{15 \times 4}{6} = 10 \). Setting the constraint for the triangular faces, consider combinations of numbers that can yield 10 for triangular faces. P... | 2 | 29882 | 360 |
If the display shows $oxed{00012}$ and we press the reciprocal key $oxed{ rac{1}{x}}$, what is the fewest number of times we must press the key to return to $oxed{00012}$? | 1. **Define the function**: Let $f(x) = \frac{1}{x}$ denote the operation of the reciprocal key on the calculator.
2. **Apply the function**: Start with 12:
\[ f(12) = \frac{1}{12} \]
3. **Apply the function again**: Now apply the function to the result from step 2:
\[ f\left(\frac{1}{12}\right) = \frac{1}{\fra... | 2 | 29883 | 245 |
The calculator shows $oxed{00010}$. After pressing the reciprocal key $oxed{ rac{1}{x}}$, what is the minimum number of times we need to press it again to see $oxed{00010}$ on the display? | 1. **Define the function**: Let $f(x) = \frac{1}{x}$ represent the operation of the reciprocal key.
2. **Apply the function**: Start with 10:
\[ f(10) = \frac{1}{10} \]
3. **Apply the function again**: Then apply the function to the result from step 2:
\[ f\left(\frac{1}{10}\right) = \frac{1}{\frac{1}{10}} = 10... | 2 | 29884 | 245 |
Currently, the calculator displays $oxed{00015}$. How many times do we need to press the reciprocal key $oxed{ rac{1}{x}}$ so that it reads $oxed{00015}$ again? | 1. **Define the function**: Let $f(x) = \frac{1}{x}$ represent the reciprocal operation.
2. **Apply the function**: Begin with 15:
\[ f(15) = \frac{1}{15} \]
3. **Apply the function again**: Now apply the function to the last result:
\[ f\left(\frac{1}{15}\right) = \frac{1}{\frac{1}{15}} = 15 \]
4. **Conclusio... | 2 | 29885 | 245 |
What is the value of $-12 + 8 \times (10 \div 2)$? | To solve the expression $-12 + 8 \times (10 \div 2)$, we follow the order of operations (PEMDAS).\n\n1. **Evaluate the expression inside the parentheses**: \n $10 \div 2 = 5$\n\n2. **Perform the multiplication**: \n $8 \times 5 = 40$\n\n3. **Perform the addition**: \n $-12 + 40 = 28$\n\nThus, the value of the ... | 28 | 29886 | 652 |
Calculate the result of $-20 + 4 \times (12 \div 3)$. | To solve the expression $-20 + 4 \times (12 \div 3)$, we again apply the order of operations (PEMDAS).\n\n1. **Evaluate the expression inside the parentheses**: \n $12 \div 3 = 4$\n\n2. **Perform the multiplication**: \n $4 \times 4 = 16$\n\n3. **Perform the addition**: \n $-20 + 16 = -4$\n\nThus, the value of... | -4 | 29887 | 652 |
Find the result of $25 - 5 \times (20 \div 4)$. | To solve the expression $25 - 5 \times (20 \div 4)$, we will follow the order of operations (PEMDAS).\n\n1. **Evaluate the expression inside the parentheses**: \n $20 \div 4 = 5$\n\n2. **Perform the multiplication**: \n $5 \times 5 = 25$\n\n3. **Perform the subtraction**: \n $25 - 25 = 0$\n\nThus, the value of... | 0 | 29888 | 652 |
800 students were surveyed about their pizza preferences: pepperoni, vegetarian, cheese, and Hawaiian. The results show that 320 students preferred pepperoni and 160 preferred vegetarian. What is the ratio of the number of students who preferred pepperoni to those who preferred vegetarian? | 1. Identify the number of students who preferred each type: 320 students preferred pepperoni and 160 preferred vegetarian. 2. Set up the ratio: The ratio is given by \( \frac{320}{160} \). 3. Simplify the ratio: To simplify \( \frac{320}{160} \), we divide both the numerator and denominator by 160: \( \frac{320 \div 16... | 2 | 29889 | 312 |
In a survey of 540 moviegoers, preferences were collected for action, drama, comedy, and horror films. If 270 preferred action and 90 preferred horror, what is the ratio of students who preferred action to those who preferred horror? | 1. Identify the number of students who preferred each film genre: 270 students preferred action and 90 preferred horror. 2. Set up the ratio: The ratio can be expressed as \( \frac{270}{90} \). 3. Simplify the ratio: To simplify \( \frac{270}{90} \), we divide both numbers by their greatest common divisor, which is 90:... | 3 | 29890 | 312 |
A poll of 500 people was conducted regarding their favorite fruits: apples, bananas, cherries, and dates. If 200 people favored apples and 150 favored bananas, what is the ratio of people who liked apples to those who liked bananas? | 1. Identify the number of people preferring each fruit: 200 people preferred apples and 150 preferred bananas. 2. Set up the ratio: The ratio is described by \( \frac{200}{150} \). 3. Simplify the ratio: To simplify \( \frac{200}{150} \), we divide by their greatest common divisor, which is 50: \( \frac{200 \div 50}{15... | 4/3 | 29891 | 312 |
A small jar of peanut butter contains 24 grams of peanut butter, while a large jar contains 480 grams. How many small jars must Maria buy to completely fill one large jar? | To determine how many small jars are needed to fill a large jar, we divide the capacity of the large jar by the capacity of the small jar.\
1. Calculate the number of small jars needed: \
\[\frac{480}{24} = 20\] \
This division yields exactly 20, meaning that 20 small jars fill the large jar completely without any r... | 20 | 29892 | 481 |
A small box can hold 45 candies, while a large box can hold 720 candies. How many small boxes are required to fully fill one large box? | To find the number of small boxes needed, we first divide the capacity of the large box by that of the small box.\
1. Calculate the number of small boxes needed: \
\[\frac{720}{45} = 16\] \
The division shows that 16 small boxes will exactly fill the large box with no candies left over.\
2. Conclusion: \
Therefor... | 16 | 29893 | 481 |
A small container can hold 50 milliliters of paint, while a large container has a capacity of 750 milliliters. Determine the minimum number of small containers required to fill the large container. | We need to find out how many small containers fit into the large one by performing a division.\
1. Calculate the required number of small containers: \
\[\frac{750}{50} = 15\] \
This calculation indicates that exactly 15 small containers are needed to fill the large container completely, as there are no remaining mi... | 15 | 29894 | 481 |
If $\log_4 x=3$, what is the value of $x$? | 1. Start with the logarithmic equation:
\[ \log_4 x = 3 \]
2. Using the definition of logarithms, we can rewrite this as:
\[ x = 4^3 \]
3. Calculate $4^3$:
\[ 4^3 = 64 \]
4. Therefore, the value of $x$ is:
\[ \boxed{64} \] | 64 | 29895 | 421 |
If $\log_2 x=5$, what is the value of $x$? | 1. Begin with the logarithmic equation:
\[ \log_2 x = 5 \]
2. Express this in exponential form:
\[ x = 2^5 \]
3. Calculate $2^5$:
\[ 2^5 = 32 \]
4. Therefore, the value of $x$ is:
\[ \boxed{32} \] | 32 | 29896 | 421 |
If $\log_3 x=4.5$, what is the value of $x$? | 1. Start with the given logarithmic equation:
\[ \log_3 x = 4.5 \]
2. Rewrite $4.5$ as a fraction:
\[ \log_3 x = \frac{9}{2} \]
3. Change to exponential form:
\[ x = 3^{\frac{9}{2}} \]
4. Simplify:
\[ 3^{\frac{9}{2}} = (3^9)^{\frac{1}{2}} = \sqrt{3^9} = \sqrt{19683} \]
5. Therefo... | 27√3 | 29897 | 421 |
A coffee cup can hold 250 milliliters of coffee, while a thermos can hold 2 liters of coffee. How many coffee cups does Naomi need to fill the thermos completely? | First, convert the thermos capacity from liters to milliliters: 2 liters is equal to 2000 milliliters.
1. **Calculate the number of coffee cups needed:**
We perform the division:
\[
\frac{2000}{250} = 8
\]
Thus, Naomi needs exactly 8 coffee cups to completely fill the thermos.
2. **Conclusion:**
Nao... | 8 | 29898 | 378 |
A small jar of peanut butter contains 450 grams, while a large jar holds 3 kilograms. If Max wants to fill the large jar with small jars, how many small jars does he need? | First, convert the large jar capacity from kilograms to grams: 3 kilograms equals 3000 grams.
1. **Calculate the number of small jars needed:**
We perform the division:
\[
\frac{3000}{450} \approx 6.6667
\]
Since we need whole jars, we round up to the nearest whole number.
2. **Round up to the nearest ... | 7 | 29899 | 378 |
A small can of paint has a capacity of 1.5 liters, while a large bucket can hold 12 liters of paint. If Lisa wants to completely fill the bucket using small cans, how many small cans does she need? | 1. **Calculate the number of small cans needed:**
We first perform the division:
\[
\frac{12}{1.5} = 8
\]
Lisa does not need to round up because 8 is already a whole number.
2. **Conclusion:**
Therefore, Lisa must buy 8 small cans to completely fill the large bucket.
So, the answer is 8. | 8 | 29900 | 378 |
The sum of the numerical coefficients in the expansion of the binomial $(x+y)^5$ is: | To find the sum of the numerical coefficients in the expansion of the binomial $(x+y)^5$, we substitute $x = 1$ and $y = 1$. This gives us the expression $(1+1)^5$. Now, we calculate: \( (1+1)^5 = 2^5 \), which equals 32. Therefore, the sum of the numerical coefficients in the expansion of $(x+y)^5$ is 32. | 32 | 29901 | 1,693 |
The sum of the numerical coefficients in the expansion of the binomial $(m+n)^4$ is: | To find the sum of the numerical coefficients in the expansion of the binomial $(m+n)^4$, we substitute $m = 1$ and $n = 1$. Therefore, we evaluate $(1+1)^4$. We find that: \( (1+1)^4 = 2^4 \), which equals 16. Thus, the sum of the numerical coefficients in the expansion of $(m+n)^4$ is 16. | 16 | 29902 | 1,693 |
The sum of the numerical coefficients in the expansion of the binomial $(p+q)^7$ is: | To find the sum of the numerical coefficients in the expansion of the binomial $(p+q)^7$, we substitute $p = 1$ and $q = 1$. As a result, we calculate $(1+1)^7$. Thus: \( (1+1)^7 = 2^7 \), which equals 128. Hence, the sum of the numerical coefficients in the expansion of $(p+q)^7$ is 128. | 128 | 29903 | 1,693 |
Maria is 5 years younger than Tom. Tom is 2 years older than Lily. Lily is 12 years old. How old is Maria? | 1. **Determine Lily's Age:**
Given that Lily is $12$ years old.
2. **Calculate Tom's Age:**
Tom is $2$ years older than Lily. Therefore, we calculate Tom's age as follows:
\[
\text{Tom's age} = \text{Lily's age} + 2 = 12 + 2 = 14 \text{ years}
\]
3. **Calculate Maria's Age:**
Maria is $5$ years youn... | 9 | 29904 | 177 |
Alex is 6 years younger than Emma. Emma is 4 years older than Brian. Brian is 10 years old. How old is Alex? | 1. **Determine Brian's Age:**
Given that Brian is $10$ years old.
2. **Calculate Emma's Age:**
Emma is $4$ years older than Brian. Therefore, we calculate Emma's age as follows:
\[
\text{Emma's age} = \text{Brian's age} + 4 = 10 + 4 = 14 \text{ years}
\]
3. **Calculate Alex's Age:**
Alex is $6$ year... | 8 | 29905 | 177 |
Nina is 3 years younger than Leo. Leo is 5 years older than Mya. Mya is 18 years old. How old is Nina? | 1. **Determine Mya's Age:**
Given that Mya is $18$ years old.
2. **Calculate Leo's Age:**
Leo is $5$ years older than Mya. Therefore, we calculate Leo's age as follows:
\[
\text{Leo's age} = \text{Mya's age} + 5 = 18 + 5 = 23 \text{ years}
\]
3. **Calculate Nina's Age:**
Nina is $3$ years younger th... | 20 | 29906 | 177 |
The graph shows the number of minutes studied by Robin (blue bar) and Taylor (red bar) over a week. On average, how many more minutes per day did Taylor study than Robin? | 1. **Calculate the daily differences**: The differences between Taylor's and Robin's study times for the week are given as $15, -5, 25, 35, -10$. \n2. **Sum the differences**: Add up these differences: $15 + (-5) + 25 + 35 + (-10) = 15 - 5 + 25 + 35 - 10 = 60$. \n3. **Count the number of days**: There are 5 days in the... | 12 | 29907 | 685 |
The chart illustrates the study durations in minutes for Mia (green bar) and Zoe (purple bar) throughout a week. How many more minutes, on average, did Zoe study compared to Mia per day? | 1. **Calculate the daily differences**: The differences are given as $20, -5, 15, 5, -10$. \n2. **Sum the differences**: $20 + (-5) + 15 + 5 + (-10) = 20 - 5 + 15 + 5 - 10 = 25$. \n3. **Count the number of days**: There are 5 days. \n4. **Calculate the average difference per day**: Average difference = \( \frac{25}{5} ... | 5 | 29908 | 685 |
The report details the minutes spent studying by Liam (orange bar) and Emma (yellow bar) during a week. Calculate the average number of additional minutes Emma studied per day compared to Liam. | 1. **Calculate the daily differences**: The daily differences are $5, 10, -15, 20, 15$. \n2. **Sum the differences**: Total difference is $5 + 10 + (-15) + 20 + 15 = 5 + 10 - 15 + 20 + 15 = 35$. \n3. **Count the number of days**: The duration is over 5 days. \n4. **Calculate the average difference per day**: Average di... | 7 | 29909 | 685 |
Alex and Jamie take turns counting starting from $1$. Alex starts and says the first number, then Jamie says the next one which is one more than what Alex said. What is the $75^{ ext{th}}$ number said by either Alex or Jamie? | 1. **Understanding the Sequence**: The numbers said by Alex and Jamie follow a simple natural counting sequence. If Alex says $1$, Jamie will say $2$, Alex will say $3$, and the pattern continues as $1, 2, 3, 4, 5, ext{...}$. Each person is just saying the next natural number in order.
2. **Identifying the Pattern**:... | 75 | 29910 | 580 |
Mia and Noah are counting together where Mia starts by saying $1$ and Noah follows by saying one more than the previous number. What is the $100^{ ext{th}}$ number that Noah will say? | 1. **Understanding the Sequence**: Mia and Noah are counting using natural numbers. Mia says $1$, Noah then says $2$, Mia says $3$, Noah says $4$, and so forth. Thus, the sequence is just continuously alternating between Mia and Noah, where each says the next number in increasing order.
2. **Identifying the Pattern**:... | 100 | 29911 | 580 |
Kara and Liam take turns counting numbers starting from $1$. Kara initiates by saying $1$, followed by Liam who says $2$. What is the $120^{ ext{th}}$ number that either of them will say? | 1. **Understanding the Sequence**: The counting sequence initiated by Kara and followed by Liam will be $1, 2, 3, 4, 5, ...$, which continues indefinitely as they take turns.
2. **Identifying the Pattern**: Each number mentioned corresponds to its position in counting natural numbers. This means that to find the $120^... | 120 | 29912 | 580 |
Jake and Molly each flip a coin, resulting in either Heads (H) or Tails (T). What is the probability that Jake's outcome is Heads while Molly's outcome is Tails? | To find the probability, we first identify the possible outcomes from both Jake and Molly flipping their coins. Each person has 2 outcomes: Heads (H) or Tails (T). Thus, the total outcomes of the two flips are given by: Total outcomes = 2 (for Jake) * 2 (for Molly) = 4. The successful outcomes where Jake's outcome is H... | 1/4 | 29913 | 811 |
Laura and Leo each draw one card from a standard deck of 52 cards. What is the probability that Laura's card is of a higher rank than Leo's card? | To solve this problem, we first consider the ranks of the cards, which are Ace (highest), King, Queen, Jack, 10 down to 2 (lowest). Each player has 52 cards available. The total number of outcomes when both draw is: Total outcomes = 52 * 52 = 2704. Next, we calculate the successful outcomes where Laura's card rank is h... | 1/41 | 29914 | 811 |
In a race, Alice and Bob each run a lap that can be measured in fractions of miles, with Alice running 0.8 miles and Bob running 0.6 miles. What is the probability that Alice runs a longer distance than Bob? | In this race, we only have two outcomes: Alice running 0.8 miles and Bob running 0.6 miles. Since we are asked to compare these two specific distances, we simply need to determine how many situations favor Alice. There is only 1 instance where Alice runs farther than Bob, as 0.8 miles is greater than 0.6 miles. The tot... | 1 | 29915 | 811 |
For a set of five distinct lines in a plane, there are exactly $M$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $M$? | To solve this problem, we analyze the maximum number of intersection points formed by five distinct lines in a plane. The maximum number of intersection points is given by the binomial coefficient $inom{5}{2} = 10$. Each pair of lines can intersect at most once, making the maximum total possible distinct intersection ... | 35 | 29916 | 320 |
For a set of three distinct lines in a plane, how many distinct points lie on two or more of the lines? What is the sum of all possible values? | To find the total number of intersection points among three lines, we consider that the maximum number of intersection points equals the number of ways to pick 2 lines from 3, or $inom{3}{2} = 3$. Each pair of lines intersects at a unique point, unless they are parallel. Thus, the potential values for the number of in... | 6 | 29917 | 320 |
In a plane, how many distinct intersection points can be formed by six distinct lines? What is the sum of all possible distinct intersection points? | To calculate the potential points of intersection for six distinct lines, we use the binomial coefficient $inom{6}{2} = 15$, suggesting that at maximum, 15 distinct intersection points can be formed if no two lines are parallel. The minimum number of intersections is 0, occurring if all lines are parallel. The valid i... | 120 | 29918 | 320 |
Vertices of a tetrahedron are to be labeled with the integers $1$ through $4$, with each integer used once. The sum of the labels on each face must be the same for all faces. How many distinct labeling arrangements are there if arrangements that can be obtained from each other through rotations are considered the same? | The total sum of the integers from $1$ to $4$ is $1 + 2 + 3 + 4 = 10$. A tetrahedron has $4$ triangular faces. Since every face contributes to the total counted sum, we divide by the number of faces to find the face sum required. Thus, for the face sums to be equal, we require:
\[ \text{Face sum} = \frac{10}{4} = 2.5, ... | 0 | 29919 | 377 |
Each vertex of a rectangular prism is to be labeled with integers from $1$ to $12$, ensuring that the sum on each face is equal for all faces. What are the total distinct arrangements possible, considering rotational symmetry? | First, calculate the total sum of integers from $1$ to $12$:
\[ 1 + 2 + 3 + \ldots + 12 = \frac{12 \times 13}{2} = 78. \]
A rectangular prism has $6$ rectangular faces. Each vertex belongs to $4$ faces. Therefore, for face sums to be equal:
\[ \text{Required Face Sum} = \frac{4 \times \text{the total sum}}{6} = \frac{4... | 0 | 29920 | 377 |
Vertices of an octahedron are to be labeled with integers from $1$ to $6$ such that each face has the same sum. How many distinct labeling arrangements can be formed, considering two arrangements are the same if one can be obtained from the other by rotation? | The sum of the integers from $1$ to $6$ is:
\[ 1 + 2 + 3 + 4 + 5 + 6 = 21. \]
An octahedron has $8$ triangular faces. So the required sum for each face must be:
\[ \frac{21}{8} = 2.625, \text{ which is again impossible with integers.} \]
This means no valid configurations can satisfy the condition of equal face sums.
... | 0 | 29921 | 377 |
$igg(2^{-3}igg)^{-2}=$ | 1. **Rewrite the expression using the property of exponents:** The property states that $(a^b)^c = a^{bc}$. Hence, we can write:
\[
\bigg(2^{-3}\bigg)^{-2} = 2^{(-3)\cdot(-2)} = 2^6
\]
2. **Calculate the power:** Now we need to calculate $2^6$:
\[
2^6 = 64
\]
3. **Conclude with the correct answer:**... | 64 | 29922 | 829 |
$igg(3^{- rac{1}{2}}igg)^{-4}=$ | 1. **Rewrite the expression using the property of exponents:** Using the property $(a^b)^c = a^{bc}$, we have:
\[
\bigg(3^{-\frac{1}{2}}\bigg)^{-4} = 3^{-\frac{1}{2}\cdot(-4)} = 3^{2}
\]
2. **Calculate the power:** We calculate $3^2$:
\[
3^2 = 9
\]
3. **Conclude with the correct answer:** Thus, the ... | 9 | 29923 | 829 |
$igg(5^{ rac{1}{3}}igg)^{-3}=$ | 1. **Rewrite the expression using the property of exponents:** Using $(a^b)^c = a^{bc}$, this can be expressed as:
\[
\bigg(5^{\frac{1}{3}}\bigg)^{-3} = 5^{\frac{1}{3}\cdot(-3)} = 5^{-1}
\]
2. **Rewrite with positive exponent:** We can write $5^{-1}$ as:
\[
5^{-1} = \frac{1}{5}
\]
3. **Conclude with... | 1/5 | 29924 | 829 |
For each positive integer $n$, let $f(n) = rac{1}{2} imes ext{sum of the first 100 integers divisible by } n. Find the largest value of $n$ for which $f(n) ext{ is less than or equal to } 2550$. | To solve the problem, we need to calculate $f(n) = rac{1}{2} imes (n + 2n + 3n + ... + 100n)$ which simplifies to $f(n) = rac{1}{2} imes n imes rac{100(100+1)}{2}$. Thus, $f(n) = 2550n$. We need $2550n \leq 2550$. Hence, $n \leq 1$. Therefore, the largest value of $n$ is $\boxed{1}$. | 1 | 29925 | 206 |
For each positive integer $n$, let $f(n) = rac{1}{3} imes ext{sum of the squares of the first } 100 \text{ integers that are multiples of } n. Find the largest value of $n$ for which $f(n) \le 10000$. | To solve this, observe that the sum of the first 100 squares is $\sum_{k=1}^{100} k^2 = \frac{100(100 + 1)(2 \cdot 100 + 1)}{6} = 338350$. Thus, $f(n) = \frac{1}{3}n^2 \cdot 338350$. We need $\frac{1}{3}n^2 \cdot 338350 \leq 10000$. Simplifying gives $n^2 \leq \frac{30000}{338350}$. This implies $n \leq 9$. Therefore, ... | 9 | 29926 | 206 |
For each positive integer $n$, let $f(n) = \sum_{k = 1}^{50} \lceil \sqrt{kn} \rceil$. Find the largest value of $n$ for which $f(n) \le 2100$. | We start by calculating $f(n) = \sum_{k=1}^{50} \lceil \sqrt{kn} \rceil$. For $n = 1$, we find $f(1) = \sum_{k=1}^{50} k = \frac{50(50+1)}{2} = 1275$. For $n = 4$, $f(4) = \sum_{k=1}^{50}\lceil 2\sqrt{k} \rceil$. Testing gives $f(4) \approx 2550$. Therefore, we should check $n$ values between 1 and 4. Testing $n = 3$, ... | 2 | 29927 | 206 |
Calculate $4^4 + 4^4 + 4^4 + 4^4 = $ | 1. **Identify the Expression**: The expression is $4^4 + 4^4 + 4^4 + 4^4$.
2. **Simplify the Expression**: There are four terms, each equal to $4^4$. Therefore, we write this as $4 imes 4^4$.
3. **Apply Exponent Rules**: Using the exponent rule $a^m imes a^n = a^{m+n}$, we simplify:
$$4 imes 4^4 = 4^1 imes ... | 4^5 | 29928 | 389 |
Find the value of $5^3 + 5^3 + 5^3 + 5^3 + 5^3 = $ | 1. **Identify the Expression**: The expression given is $5^3 + 5^3 + 5^3 + 5^3 + 5^3$.
2. **Simplify the Expression**: There are five terms of $5^3$, which can be written as $5 imes 5^3$.
3. **Apply Exponent Rules**: We can further simplify it using the exponent rule:
$$5 imes 5^3 = 5^1 imes 5^3 = 5^{1+3} = ... | 5^4 | 29929 | 389 |
Evaluate the expression $3^5 + 3^5 + 3^5 + 3^5 + 3^5 + 3^5 + 3^5 = $ | 1. **Identify the Expression**: The given expression is $3^5 + 3^5 + 3^5 + 3^5 + 3^5 + 3^5 + 3^5$.
2. **Simplify the Expression**: Since there are seven terms each equal to $3^5$, we can rewrite it as $7 imes 3^5$.
3. **Apply Exponent Rules**: To express $7 imes 3^5$ in exponent form, we note that this does not y... | 7 * 3^5 | 29930 | 389 |
The digits 0, 1, 5, 6, and 8 are each used once to form the smallest possible even five-digit number. The digit in the tens place is | 1. The number must end with an even digit. The available even digits are 0, 6, and 8.\n\n2. To form the smallest number, we need to consider the available even digits. Here, 0 cannot be in the unit position as it would not create a five-digit number. Thus, we choose the smallest available even digit, which is 6.\n\n3. ... | 8 | 29931 | 439 |
The digits 2, 3, 4, 5, and 8 are each used once to form the smallest possible even five-digit number. The digit in the tens place is | 1. To form the smallest even five-digit number, the number must end with an even digit. The available even digits are 2 and 4, and 8.\n\n2. Among these, 2 is the smallest, but placing it in the units position may not yield the smallest overall number. Hence, choose 4 as the units digit to allow smaller digits in higher... | 8 | 29932 | 439 |
The digits 1, 3, 5, 6, and 7 are each used once to form the smallest possible even five-digit number. The digit in the tens place is | 1. Identify the even digits available from 1, 3, 5, 6, and 7. Here, the only even digit is 6.\n\n2. To form the smallest possible number, we need to place 6 at the units position, ensuring the number remains even.\n\n3. For the ten-thousands place, the smallest number left is 1, which we place there.\n\n4. The next sma... | 7 | 29933 | 439 |
If $y ext{ is a positive real number, then } rac{y^2}{ rac{1}{y}} = ?$ | 1. **Understanding the expression**: We start with the expression $\frac{y^2}{\frac{1}{y}}$. This involves a fraction where the denominator is a reciprocal.\n\n2. **Simplifying the fraction**: According to the rules of dividing by a fraction, we have $\frac{y^2}{\frac{1}{y}} = y^2 \cdot y = y^{2 + 1} = y^3$.\n\n3. **Fi... | 3 | 29934 | 188 |
If $z \geq 0$, then $\sqrt{z^2 \sqrt{z}} = ?$ | 1. **Understanding the expression**: Start with the expression $\sqrt{z^2 \sqrt{z}}$. This has a square root of a product. \n\n2. **Simplifying the innermost square root**: We start by simplifying $\sqrt{z}$ to $z^{\frac{1}{2}}$.\n\n3. **Substituting into the expression**: Now substitute back: $\sqrt{z^2 \cdot z^{\frac... | 5 | 29935 | 188 |
If $w \geq 0$, then $\sqrt{w \sqrt{w \sqrt{w^3}}} = ?$ | 1. **Understanding the expression**: Consider the expression $\sqrt{w \sqrt{w \sqrt{w^3}}}$. This has multiple layers of square roots and powers. \n\n2. **Simplifying the innermost square root**: Start with the innermost term. For $\sqrt{w^3} = w^{\frac{3}{2}}$.\n\n3. **Substituting up**: Substitute $w^{\frac{3}{2}}$ b... | 5 | 29936 | 188 |
Let $b_n$ be the number of permutations $(y_1, y_2, imes, y_n)$ of the numbers $(1, 2, imes, n)$ such that the $n$ ratios $\frac{y_k}{k}$ for $1 \leq k \leq n$ are all distinct. Prove that $b_n$ is even for all $n \geq 2.$ | To show that $b_n$ is even for all $n \geq 2$, consider an involution defined similarly to the previous problem. Take a permutation $(y_1, imes, y_n)$ and define the involution $(z_1, z_2, \ldots, z_n)$ such that $z_k = y_{n+1-y_k}$ for all $k$. Now, notice that for $n \geq 2$, this mapping creates pairs unless we app... | 0 | 29937 | 195 |
Let $c_n$ be defined as the number of permutations $(m_1, m_2, \ldots, m_n)$ of the numbers $(1, 2, \ldots, n)$ such that the ratios $\frac{m_k}{k}$ for each $k$ from $1$ to $n$ are distinct. Show that $c_n$ is odd for all $n \geq 1$. | We will approach this problem using a similar involution argument. Define the mapping $(p_1, p_2, \ldots, p_n)$ such that $p_k = m_{m_k}$. An important observation here is that if $m_k \neq k$, then there will be a pair $(p_k, p_{p_k})$ that does permute, confirming it is an involution. Similar to our initial example, ... | 1 | 29938 | 195 |
Let $d_n$ represent the number of arrangements $(z_1, z_2, \ldots, z_n)$ of numbers $(1, 2, \ldots, n)$ such that the expressions $\frac{z_k}{k}$ for $1 \leq k \leq n$ remain distinct. Demonstrate that $d_n$ is even for every integer $n \geq 2$. | For any arrangement of $(z_1, z_2, \ldots, z_n)$, we can again use a similar involution mapping: define the mapping such that $w_k = z_{n+1-z_k}$. For $n \geq 2$, each permutation will create matching pairs unless we have a repeated value, such as in the identity case which does not contribute to the overall count of v... | 0 | 29939 | 195 |
A marathon took a total of 4 hours and 30 minutes to complete. How many minutes is this? | To find the total duration of the marathon in minutes, we need to convert the hours into minutes and then add the remaining minutes.
1. **Convert hours to minutes**:
- For 4 hours, the total minutes are:
4 ext{ hours} \times 60 ext{ minutes/hour} = 240 ext{ minutes}
2. **Add the extra minutes**:
... | 270 | 29940 | 1,077 |
A concert lasted for 2 hours and 45 minutes. How many minutes did the concert last? | To find out how many minutes the concert lasted, we need to convert the hours into minutes first:
1. **Convert hours to minutes**:
- For 2 hours, the total minutes are:
2 ext{ hours} \times 60 ext{ minutes/hour} = 120 ext{ minutes}
2. **Add the extra minutes**:
- The concert also lasted an additi... | 165 | 29941 | 1,077 |
A lecture took place over a period of 3 hours and 20 minutes. What is the total duration in minutes? | To convert the duration of the lecture into minutes, we proceed as follows:
1. **Convert hours to minutes**:
- For 3 hours, the total minutes are:
3 ext{ hours} \times 60 ext{ minutes/hour} = 180 ext{ minutes}
2. **Add the extra minutes**:
- The lecture also lasted an additional 20 minutes.
- ... | 200 | 29942 | 1,077 |
There are $150$ seats in a theater. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone? | To solve the problem, we need to arrange the seating to maximize empty seats while ensuring the next person has to sit next to someone.
1. **Understanding the pattern**: Similar to the previous problem, if we alternate seated persons and empty seats, we can maximize the number of empty seats. The arrangement forms as ... | 75 | 29943 | 168 |
In a bus station, there are $200$ seats in a waiting area. What is the fewest number of seats that must be occupied so that every newly arriving person must sit next to someone? | We need to find a way to sit as many people as possible while still ensuring the next person has to sit next to someone already seated.
1. **Understanding the arrangement**: We can employ the pattern of sitting (occupied) and leaving an empty seat, represented as $ ext{p o p ...}$.
2. **Counting the groups**: Each gr... | 100 | 29944 | 168 |
A sports arena has $180$ seats in a row. What is the fewest number of seats that must be filled so that the next person to arrive must sit next to someone? | To maximize empty seats while ensuring that no new person can sit down without being next to someone already seated:
1. **Understanding the seating arrangement**: We can alternate seating individuals with empty seats, as in $ ext{p o p ...}$.
2. **Counting the groups**: Each 'p o' arrangement takes up 2 seats (one oc... | 90 | 29945 | 168 |
How many primes less than $50$ have $3$ as the ones digit? (Assume the usual base ten representation) | 1. **Identify the numbers with 3 as the ones digit under 50**: The numbers are $3, 13, 23, 33, 43$.
2. **Check each number for primality**:
- $3$ is prime.
- $13$ is prime.
- $23$ is prime.
- $33$ is not prime (divisible by $3$).
- $43$ is prime.
3. **Count the prime numbers**: The prime nu... | 4 | 29946 | 611 |
How many primes less than $30$ have $1$ as the ones digit? (Assume the usual base ten representation) | 1. **Identify the numbers with 1 as the ones digit under 30**: The numbers are $1, 11, 21$.
2. **Check each number for primality**:
- $1$ is not prime.
- $11$ is prime.
- $21$ is not prime (divisible by $3$).
3. **Count the prime numbers**: The only prime number in the list is $11$. There is 1 such n... | 1 | 29947 | 611 |
How many primes less than $60$ have $9$ as the ones digit? (Assume the usual base ten representation) | 1. **Identify the numbers with 9 as the ones digit under 60**: The numbers are $9, 19, 29, 39, 49, 59$.
2. **Check each number for primality**:
- $9$ is not prime (divisible by $3$).
- $19$ is prime.
- $29$ is prime.
- $39$ is not prime (divisible by $3$).
- $49$ is not prime (divisible by $7... | 3 | 29948 | 611 |
In a triangle, two sides measure 7 cm and 9 cm, with the angle between them measuring 60 degrees. What is the area of the triangle? | To calculate the area of the triangle, we can use the formula: Area = 1/2 * a * b * sin(C), where 'a' and 'b' are the lengths of the two sides and 'C' is the included angle. Here, a = 7 cm, b = 9 cm, and C = 60 degrees.
1. Plug in the values: Area = 1/2 * 7 * 9 * sin(60 degrees).
2. Calculate sin(60 degrees): sin(60 d... | 27.31 | 29949 | 814 |
A rectangle has a length of 10 meters and a width of 5 meters. Calculate the perimeter of the rectangle. | The perimeter P of a rectangle can be calculated using the formula: P = 2(l + w), where 'l' is the length and 'w' is the width.
1. Given l = 10 m and w = 5 m.
2. Substitute the values into the formula: P = 2(10 + 5).
3. Calculate the sum: P = 2(15).
4. Now multiply to find the perimeter: P = 30 m.
Thus, the perimeter ... | 30 | 29950 | 814 |
In a regular pentagon, each side measures 6 cm. What is the total perimeter of the pentagon? | The perimeter 'P' of a polygon can be found by multiplying the length of one side 's' by the number of sides 'n': P = n * s.
1. A regular pentagon has 5 sides, so n = 5.
2. Given the side length s = 6 cm, we substitute the values: P = 5 * 6.
3. Calculate the perimeter: P = 30 cm.
Thus, the total perimeter of the pent... | 30 | 29951 | 814 |
Consider the pattern created by vertical stripes and alternating semicircles above and below a straight line. Each stripe spans vertically between the semicircles. How many of the following transformations will leave the figure unchanged? (1) Some rotation around a point on the line of symmetry, (2) Some translation al... | 1. **Some rotation around a point on the line of symmetry:** A rotation of 180 degrees about the midpoint of a section of the pattern aligns the semicircles above with those below, thus mapping the figure onto itself. \n2. **Some translation along the direction of the stripes:** Translating the figure by the width of o... | 3 | 29952 | 751 |
A pattern consists of alternating triangles facing left and right along a horizontal line. Determine how many transformations will preserve this figure: (1) Rotation about a vertex of a triangle, (2) Translation along the horizontal line, (3) Reflection across the horizontal line, (4) Reflection across a vertical line. | 1. **Rotation about a vertex of a triangle:** This transformation does not preserve the orientation of the triangles; a rotation of 180 degrees will swap the triangles’ positions relative to their orientation. \n2. **Translation along the horizontal line:** Translating the pattern along the horizontal line by the dista... | 1 | 29953 | 751 |
A design consists of concentric squares and diagonal lines extending from the corners. How many transformations can maintain this design: (1) Rotation around the center of the squares, (2) Translation upwards, (3) Reflection across a diagonal line, (4) Reflection across a horizontal line? | 1. **Rotation around the center of the squares:** A 90-degree rotation (or multiples) of the squares will keep the design in its original alignment since all squares are symmetric. \n2. **Translation upwards:** This transformation does not maintain the design because the squares would move away from the center and no l... | 1 | 29954 | 751 |
Given a stack map with heights represented as 3, 2, 4, and 1 in four vertical columns from left to right, what would the front view look like? | To find the front view of the columns represented by the stack map, we identify the maximum heights from each vertical stack as follows:
1. **Identify the tallest stack in each column:**
- **First column:** Represents 3 cubes.
- **Second column:** Represents 2 cubes.
- **Third column:** Represents 4 cubes.
... | 4 | 29955 | 49 |
A stack map shows the heights of cubes in four vertical columns: 5, 2, 3, and 6. What is the height of the stacks when viewed from the front? | To determine the front view of the stack map, we need to find the tallest stack in each column:
1. **Identify the tallest stack in each column:**
- **First column:** Has a height of 5 cubes.
- **Second column:** Has a height of 2 cubes.
- **Third column:** Has a height of 3 cubes.
- **Fourth column:** Has ... | 6 | 29956 | 49 |
In a stack map, the heights of cubes in three columns are noted as 2, 4, and 3. What does the front view reveal? | To analyze the front view based on the stack map:
1. **Identify the tallest stack in each column:**
- **First column:** Has 2 cubes.
- **Second column:** Has 4 cubes.
- **Third column:** Has 3 cubes.
2. **Construct the front view:**
- Listing the tallest stacks gives us 2 cubes for the first column, 4 cub... | 4 | 29957 | 49 |
Mia is 5 years younger than David. David is 4 years older than Sarah. Sarah is 12 years old. How old is Mia? | 1. **Determine Sarah's Age:** Given that Sarah is 12 years old.
2. **Calculate David's Age:** David is 4 years older than Sarah. Therefore, we calculate David's age as follows:
\[
\text{David's age} = \text{Sarah's age} + 4 = 12 + 4 = 16 \text{ years}
\]
3. **Calculate Mia's Age:** Mia is 5 years younger... | 11 | 29958 | 173 |
Liam is 6 years younger than Emma. Emma is 5 years older than Noah, who is 10 years old. How old is Liam? | 1. **Determine Noah's Age:** Given that Noah is 10 years old.
2. **Calculate Emma's Age:** Emma is 5 years older than Noah. Therefore, we calculate Emma's age as follows:
\[
\text{Emma's age} = \text{Noah's age} + 5 = 10 + 5 = 15 \text{ years}
\]
3. **Calculate Liam's Age:** Liam is 6 years younger than ... | 9 | 29959 | 173 |
Ella is 3 years older than Jake. Jake is 2 years younger than Lily, who is 18 years old. How old is Ella? | 1. **Determine Lily's Age:** Given that Lily is 18 years old.
2. **Calculate Jake's Age:** Jake is 2 years younger than Lily. Therefore, we calculate Jake's age as follows:
\[
\text{Jake's age} = \text{Lily's age} - 2 = 18 - 2 = 16 \text{ years}
\]
3. **Calculate Ella's Age:** Ella is 3 years older than ... | 19 | 29960 | 173 |
Mia wants to create a scaled-down version of the Eiffel Tower, which is 300 meters tall. The original Eiffel Tower can support 40,000 kg. Mia's model only needs to support 0.025 kg. How tall, in meters, should Mia's tower be? | 1. **Find the ratio of the weights between the original tower and the model**: The actual Eiffel Tower supports 40,000 kg while Mia's model supports 0.025 kg. The ratio is:
\[\frac{40000 \text{ kg}}{0.025 \text{ kg}} = 1600000\]
2. **Determine the scale of the model**: The weights scale as the cube of the height, s... | 7.5 | 29961 | 194 |
Jack is building a scale model of a lighthouse that stands 50 meters tall and has a capacity of 60,000 liters of fuel. His small model will only hold 0.02 liters. How tall, in meters, should Jack's lighthouse model be? | 1. **Calculate the volume ratio of the lighthouse and the model**: The actual lighthouse holds 60,000 liters and Jack's model holds 0.02 liters, giving:
\[\frac{60000 \text{ liters}}{0.02 \text{ liters}} = 3000000\]
2. **Relate this volume ratio to height**: The volume ratio relates to height in cubic form, thus:
... | 0.34 | 29962 | 194 |
Sophia is making a model of a skyscraper that is 90 meters tall and occupies 80,000 cubic meters of space. Her model only has a space of 0.125 cubic meters. What should be the height of Sophia's skyscraper model in meters? | 1. **Find the volume ratio between the skyscraper and the model**: The actual skyscraper occupies 80,000 cubic meters and Sophia's model occupies 0.125 cubic meters, therefore:
\[\frac{80000 \text{ m}^3}{0.125 \text{ m}^3} = 640000\]
2. **Relate this ratio to the heights**: The ratio of the heights is the cube root... | 2.25 | 29963 | 194 |
A delivery truck has six wheels (five driving wheels and one spare). After driving for $60,000$ miles, how many miles did each wheel effectively cover? | 1. **Total Miles Driven by All Wheels**: The truck has six wheels, but only five are used at any moment for driving. Therefore, the total number of tire-miles over the course of $60,000$ miles is calculated by multiplying the total miles driven by the number of driving wheels:
$60,000 ext{ miles} \times 5 = 3... | 50000 | 29964 | 340 |
A bus travels $45,000$ miles and has eight different wheels (seven operational wheels and one spare). How many miles does each wheel end up being used? | 1. **Total Miles Driven by All Wheels**: The bus has eight wheels, with only seven wheels functioning at any time. Thus, the total number of tire-miles when the bus has traveled $45,000$ miles is given by:
$45,000 \text{ miles} \times 7 = 315,000 \text{ tire-miles}$
2. **Equal Usage Among Wheels**: To fi... | 39375 | 29965 | 340 |
A sports car with five tires (four for driving and one spare) drove a total of $50,000$ miles. How many miles did each tire effectively cover? | 1. **Total Miles Driven by All Tires**: The sports car has five tires, but only four are utilized for driving at a time. The total tire-miles, given the car's mileage, is calculated as follows:
$50,000 \text{ miles} \times 4 = 200,000 \text{ tire-miles}$
2. **Equal Usage Among Tires**: Since the problem ... | 40000 | 29966 | 340 |
A new shipping rate charges $25$ cents for the first pound and $18$ cents for each additional pound or fraction thereof. What is the cost for a package weighing $3.2$ pounds? | 1. **Identify base rate and additional rate:** The postage rate is $25$ cents for the first pound and $18$ cents for each additional pound or fraction thereof.\n\n2. **Calculate the total weight beyond the first pound:** The package weighs $3.2$ pounds. Subtracting the first pound gives $3.2 - 1 = 2.2$ pounds remaining... | 0.79 | 29967 | 116 |
A local courier charges $40$ cents for the first ounce and $25$ cents for each additional ounce or fraction. What would be the postage for a parcel weighing $6.7$ ounces? | 1. **Identify the rates:** The first ounce costs $40$ cents, and each additional ounce costs $25$ cents.\n\n2. **Calculate the weight beyond the first ounce:** For a $6.7$ ounce parcel, $6.7 - 1 = 5.7$ ounces remain.\n\n3. **Determine additional charges:** Rounding up $5.7$ ounces leads to $6$ additional charges (as we... | 1.90 | 29968 | 116 |
For a weight-based delivery service, the cost is $50$ cents for the first pound and $30$ cents for every additional pound or part thereof. How much would it cost to ship a $5.3$ pound box? | 1. **Identify the rates:** The postage charge is $50$ cents for the first pound plus $30$ cents for each additional pound or part thereof.\n\n2. **Calculate weight beyond the first pound:** The box weighs $5.3$ pounds. This means $5.3 - 1 = 4.3$ pounds are additional.\n\n3. **Determine additional charges:** Rounding $4... | 2.00 | 29969 | 116 |
If $\log_8 x = 4$, what is the value of $x$?
\textbf{(A)}\ 256 \qquad \textbf{(B)}\ 2048 \qquad \textbf{(C)}\ 512 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ \text{none of these} | 1. Start with the logarithmic equation:
$$\log_8 x = 4$$
2. This means that:
$$x = 8^4$$
3. We can evaluate $8^4$. Knowing that $8 = 2^3$, we have:
$$x = (2^3)^4 = 2^{12}$$
4. Calculate $2^{12}$:
$$2^{12} = 4096$$
5. Therefore, the value of $x$ is:
$$\boxed{4096 \quad \textbf{(E)}}$$ | 4096 | 29970 | 470 |
If $\log_5 x = 3.5$, determine the value of $x$.
\textbf{(A)}\ 200 \qquad \textbf{(B)}\ 125 \qquad \textbf{(C)}\ 3125 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ \text{none of these} | 1. Begin with the logarithmic equation:
$$\log_5 x = 3.5$$
2. Rewrite $3.5$ as a fraction:
$$\log_5 x = \frac{7}{2}$$
3. Rewriting in exponential form gives us:
$$x = 5^{\frac{7}{2}}$$
4. This can be simplified to:
$$x = \sqrt{5^7} = 5^3 \cdot \sqrt{5}$$
5. We calculate $5^3$:
$$5^3 = 125$$
6. ... | 125√5 | 29971 | 470 |
If $\log_4 x = 3$, find the value of $x$.
\textbf{(A)}\ 64 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 256 \qquad \textbf{(E)}\ \text{none of these} | 1. Start from the logarithmic equation:
$$\log_4 x = 3$$
2. By converting to exponential form:
$$x = 4^3$$
3. Now we can calculate $4^3$:
$$4^3 = (2^2)^3 = 2^{2\cdot3} = 2^6$$
4. Calculate $2^6$:
$$2^6 = 64$$
5. Therefore, the value of $x$ is:
$$\boxed{64 \quad \textbf{(A)}}$$ | 64 | 29972 | 470 |
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