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How do you simplify i^21 + i^30?
1 Answer
Nov 13, 2015
$i - 1$
Explanation:
Observe that the powers of $i$ are cyclic:
${i}^{0} = 1$
${i}^{1} = i$
${i}^{2} = - 1$
${i}^{3} = - i$
${i}^{4} = 1$
$\ldots$
So, to find high powers of $i$, we can calculate the modulus $4$ Since $21 = 5 \cdot 4 + 1$, and $30 = 7 \cdot 4... | {} |
# Does either Quantum Field Theory or the Standard Model of Particle Physics predict the maximum number of particles or fields that can exist?
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### Are calculus and differential geometry required for building neural networks?
Neural networks are essentially just repeated matrix multiplications and applications of an activation function, so you really don't need a great deal of linear algebra to construct a simple neural ...
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# Extremely ($90$%) biased coins. What information can we derive/assume based on results of only $10$ coin flips?
Let's suppose there are $2$ heavily biased coins such that coin A has a bias of coming up $90$% heads and coin B has a bias of coming up $90$% tails. Both coins are placed in a bag and one is randomly chos... | {} |
# balancing redox reactions half reaction method
In the above equation, there are $$14 \: \ce{H}$$, $$6 \: \ce{Fe}$$, $$2 \: \ce{Cr}$$, and $$7 \: \ce{O}$$ on both sides. not have to do anything. The OH- ions, Then, on that side of the equation which contains both (OH. This is called the half-reaction method of balanc... | {} |
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### A probability question
Two analysts make prediction of the movement of a stock separately. Analyst A have an accuracy of $p_1$, which means his prediction is correct at the probability of $p_1$, whi... | {} |
## Abstract
This study empirically investigates the relationship between retirement duration and cognition among older Irish women using microdata collected in the third wave of The Irish Longitudinal Study on Ageing. Ordinary least squares (OLS) regression estimates indicate that the longer an individual has been ret... | {} |
# Generate the shortest De Bruijn
A De Bruijn sequence is interesting: It is the shortest, cyclic sequence that contains all possible sequences of a given alphabet of a given length. For example, if we were considering the alphabet A,B,C and a length of 3, a possible output is:
AAABBBCCCABCACCBBAACBCBABAC
You will ... | {} |
#### Volume 12, issue 2 (2012)
Recent Issues
The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear Contacts ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Free group automorphisms with parabolic boundary ... | {} |
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I have a very long diatribe on the state of document editors that I am working on, and almost anyone that knows me has heard parts of it, as I frequently rail against all of them: MS Word, Google Docs, Dropbox Paper, etc. In the process of writing up my screed I started thinking what a proper do... | {} |
# Chapter 16 - Questions and Problems - Page 771: 16.29
$[H_3O^+] = 2.51 \times 10^{- 5}M$
#### Work Step by Step
1. Calculate the pH. pOH + pH = 14 9.4 + pH = 14 pH = 4.6 2. Find $[H_3O^+]$ $[H_3O^+] = 10^{- 4.6}$ $[H_3O^+] = 2.51 \times 10^{- 5}$
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# 11: Chi-Square and ANOVA Tests
This chapter presents material on three more hypothesis tests. One is used to determine significant relationship between two qualitative variables, the second is used to determine if the sample data has a particular distribution, and the last is used to determine ... | {} |
+0
# precalc
0
44
2
+4
Can someone help me with this very complex question? I don't even know where to start.
Jan 27, 2023
#1
0
Using row properties of matrices:
$$\mathbf{A} \mathbf{u} = (2, 6, -2)$$
$$\mathbf{A} \mathbf{v} = (-2, 8, 12)$$
Jan 27, 2023
#2
0
Hi!
In these type of problems, you should start by... | {} |
# Analysis of Discrete Data #1 – Overview
Posted on September 11, 2014 by
This lesson is an overview of the course content as well as a review of some advanced statistical concepts involving discrete random variables and distributions, relevant for STAT 504 — Analysis of Discrete Data. This Lesson assumes that you ha... | {} |
# Is ESG a Factor?
July 2020
Read Time: 20 min
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Key Points
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I need to prove the following predicate logic sequent using natural deduction:
$\exists y \forall x (P(x) \rightarrow x = y) \vdash \forall x \forall y (P(x) \land P(y) \rightarrow x = y)$
This is my half-finished proof. I hope I'm on the right track but there is something abo... | {} |
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https://huggingface.co/datasets/open-web-math/open-web-math
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