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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How many times larger is $ 8.19\cdot10^{-14} $ than $10^{-19} $

I'm reading in a book: While the rest energy contained in an electron, $ 8.19\cdot10^{-14} J $, may seem ridiculously small, on the atomic scale it is tremendous. Compare our answer to $10^{-19} J $, the energy gained or lost by a single atom…
Jon
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GRE Problem: Square of square root of a negative number

I am solving a problem from a GRE guide, and stuck at the following problm. Given that $-1 < a < 0 < \left| a \right| < b < 1$ which of the following quantity is greater? $$\left(\frac{a^2 \sqrt{b}}{\sqrt{a}}\right)^2$$ or $$\frac{a…
Ali Baig
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Is this true? $e^{ab^{2}} = e^{a^{b^{2}}} = e^{2ab}$

Is $$e^{ab^{2}} = e^{a^{b^{2}}} = e^{2ab}?$$ I'm only really curious to know if the first term equals the second term, I just wanted to show my steps.
Jack Pan
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When it comes to the laws of powers, is there a specific order?

(I see an error I made, but I'd still like to know if there is a specific order.) I have here $\left(6^{-36}\right)/\left(6^{-16}\right)\cdot\left(6^{16}\right)$. If I do the division first, it's $-36$ minus $-16$, making an addition of plus $16$…
Dialectics
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Asymmetry in Exponentiation

So for addition, there are two operations, addition and its inverse, subtraction. \begin{equation} x + y = z \end{equation} \begin{equation} y + x = z \end{equation} \begin{equation} z - y = x \end{equation} \begin{equation} z - x =…
Joshiepillow
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Factorise: $a^5+a^4+a^3+a^2+a+ 1$

Honestly I have no idea where to start with this one If i take out the 'a' first, where do i go after that Can you think of any other ways to go about this question?
mhm
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Need to work out a formula with exponential growth.

Lets say I start with 10 gold, my income is 15 per hour. I can increase my income with 15 by purchasing an upgrade that costs 30 (as many times as I want). If i expand this into spreadsheets i can calculate it, but I haven't been able to make a…
Rachael Zacc
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Is it true that $(a+b)^p≥a^p+b^p$ for $p>1$

Let $a,b\in\mathbb R_{\geq 0}$ and let $p>1$. Is it true that $$ (a+b)^p\geq a^p+b^p $$ holds? It's obvious for $p\in\mathbb N_{>1}$. I was thinking of making an argument involving derivatives. So for each $p\in\mathbb N_{>1}$ we have that…
Sha Vuklia
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What's wrong with this reasoning? $((-2)^2)^\frac{1}{2} = (-2)^1 = -2$ vs $((-2)^2)^\frac{1}{2} = 4^\frac{1}{2} = 2$

We can, although not in very formal way, prove the property $((x)^b)^a$=$(x)^{ba}$ this way: $((x)^a)^b=(x)^a*(x)^a*…*(x)^a$ which is a product of $b$ number of $(x)^a$…
ToTheSpace 2
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exponentiation question

I have a homework question on exponents, the question asks "simplify the expression and eliminate any negative exponents" The Question is as follows $$(2x^2y^4)^3(3x^{-3}y)^2$$ and my working…
Ben
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Why raise an optimization parameter to the power of two?

I've read this SO thread which doesn't answer my question. So I have a cost function to minimize: where R is the given outline of a rectangle, and R* is the actual one. Why is it raising to the power of 2 for this option? For example for S (area),…
user3132457
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How to derive $e^x = \lim_{n \to \infty} \left( 1 + \frac x n \right)^ n$ directly from $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^ n$?

If given the definition of $ e $: $$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^ n$$ Using this fact alone, can it directly derive $$e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n$$ for positive and negative integer $x$ and in…
nonopolarity
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Is it safe to assume that $x=y$ for the equation $x^y=y^x$?

This thought came to me when I was thinking about exponents. Is the only solution for $x^y=y^z$ be $x=y$? How exactly would we prove there are no other numbers or how would we exactly solve this?
Physics_Learner
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Dividing exponent with same base

There are $~2^{80}~$ possibilities to calculate and I want to divide it by $~2~$ to process it by two computers at the same time to find the answer maybe sooner. How can I divide $~2^{80}~$ by $~2~$?
R1w
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Given $ax^n = 2^k$ is it possible to calculate $k$ if $a$, $x$ and $n$ are known

For example, we know that: $2.143483648 * 10^9 = 2^{31}$ If we were given: $2.143483648 * 10^9 = 2^k$ How do we find $k$ ?
j b
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