Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [exponentiation]

Questions about exponentiation, the operation of raising a base $b$ to an exponent $a$ to give $b^a$.

Exponentiation is a mathematical operation which produces a power $a^n$ from a base $a$ and an exponent $n$. The objects involved are usually numbers, but the procedure can be generalized to matrices, elements in algebraic structures, sets, etc.

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What is the difference between squaring a negative number inside and outside of parentheses?

What is the difference between: $ -3^2 $ and $ (-3)^2 $ ? I know $-3^2 = -9$ and $(-3)^2 = 9$ but I don't know why.
Anon
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Solution for an exponential expression without using logarithms, with two defined variables

If $60^a=3$ and $60^b=5$, what is the result of $12^{\frac{1-a-b}{-2-2b}}$? This has to be done without logarithms. The past four hours were helpless to me. Any hint, solution is welcome, I just want to learn it, or it will continue to bug my…
swhat7
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Same first digit

We raise the numbers 2 and 5 to the same positive integer power and get two numbers that have the same first (leftmost) digit. What are the possible values of the first digit?
Samuel
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Why is $\sqrt x\ne\frac1{x^2}$?

Which step is illegal? $\sqrt x=x^{\frac12}$ $\frac12=2^{-1}$ $x^{\frac12}= x^{2^{-1}}$ $a^{b^c}=a^{bc}$ $x^{2^{-1}}=x^{-2}$ $a^{-b}=\frac1{a^b}$ $x^{-2}=\frac1{x^2}$ Am I correct in assuming that it is step 3, because $a^{b^c}$ is not the same as…
Adam Lee
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Generalise $y = pb^k$ where $k = c(1-x^a)$ given values for $p$, $b^c$ and $a$

I'm working on a pricing algorithm for work and I've reached a point where the rust on my brain is blocking the creative juices. I did a maths degree years ago, obviously deserving the odd fails I recorded, as I can't remember the rules for…
e_i_pi
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Find the remainder of $128^{1000}/153$.

I have tried for almost more than an hour to find the remainder of the following $$\frac{128^{1000}}{153}$$ I applied remainder theorem to get the answer but could not succeed. Any suggestion or help is appreciated. Thanks.
Rain
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If $x^{x^x} = 2^{-\sqrt{2}}$, find $x^{-2}$

I've been struggling with this one today. Applying the logarithmic function to both sides of the equation doesn't seem to work. Any ideas?
Rodrigo Almeida
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Tesla Powerwall break even interest rate

I just read this article on Reddit yesterday, and I'm trying to figure out if it makes financial sense to invest in this thing. Assuming the system costs \$16,790 and saves $2110/yr, at what interest rate does it become better to leave your money in…
mpen
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Exponential vs Polynomial growth

May somebody give me a prove why $n^{r} < c^{n}$ when $r \in R_{>0}$, $ c \in R_{>1}$ and $ n \in N$ (for all $n$ at some point).
get rekt m8
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How to solve this exponential equation

Given this equation $$5^m - 7^m = -2$$ It is clear that the $m = 1$ but how do we solve it?
Samuel
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Simplifying $2^{\log_2(n)/2}$

I'm just completely blanking out right now. So can someone please explain to me how $2^{\log_2(n)/2}$ simplifies to $\sqrt{n}$?
Jasmine
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Odd powers with a negative base

In our new books in freshman highschool,it is mentioned that : When $a<0$ then, the expression $$a^{m \over n} m,n\in \mathbb Z $$ even if the $n$ in the denominator is odd. And my teacher gave the argument that: $$((-8)^2)^{1 \over 6}=64^{1 \over…
Logan Luther
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Restriction when use $a^{p/q}=(a^{1/q})^{p}$

What's the restriction (for $a, p, q$) when I use the following in proof? $a^{p/q}=(a^{1/q})^{p}$ The calculus book give a list said: $a$ $real$, $q$ $odd$ --------------- $a^{1/q}$, called the $q$th root of $a$, is the number $b$ such that…
linear_combinatori_probabi
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Rewriting 4^(n) as a^(n-1)

How can I rewrite $4^n$ as $a^{n-1}$? Where $a$ is known? I thought about solving with logarithms but that seems long-winded for something so simple... I think I am just too tired right now, I cannot think. $$ 4^n=a^{n-1} \implies n \log 4 = (n-1)…
minseong
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Formula to Find the Exponent

Is there a formula that determinants the exponent used to get to an answer? n^x = y x = ? n and y are known ex: 2^x = 128 how do we get to x = 7?
ToMakPo
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