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.

4326 questions
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$x^x = y$. Given $y$, find $x$.

Title is fairly self-explanatory. For example, for $y=27$, $x$ would be $3$. Specifically I was trying to find $x$ given $y = 10^{100}$, but I'd like to know how to solve it for any value of $y$.
Timothy Smith
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If $a^{p}\cdot b^{p}= (a\cdot b)^{p}$ then why $-1^{2}\cdot 3^{2}\neq (-1\cdot 3)^{2}$

If $a^{p}\cdot b^{p}= (a\cdot b)^{p}$ then why $$-1^{2}\cdot 3^{2}\neq (-1\cdot 3)^{2}\\ -1\cdot 9\neq (-3)^{2}\\ -9\neq 9$$ I'm sorry, I don't know how to put latex code.
Jose
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List the numbers in order

How would I list these numbers in order without using a calculator? Thank you List these numbers in increasing order: $2^{800}$, $3^{600}$, $5^{400}$, $6^{200}$
Maximiliano
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what are the consequences of using ${-0}$ in exponentiation?

mainly in regards to $10^{-0}$, $-10^{-0}$, $0^{-0}$, $-0^0$ and $-0^{-0}$ i have have tried wolframalpha but i can't get it to interpret ${-0}$ as ${-0}$, it just interprets it as ${0}$
Filly-linni
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Simplifcation of $e^{(-3 \pi i)/4}$

I am trying to figure out how my textbook simplifies this: $$\frac{1}{4}e^{(-3 \pi i)/4} = -\frac{1}{4}e^{(\pi i)/4}$$ I know that $e^{\pi i} = -1$ but what is going on here?
Jwan622
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What is the fundamental error in my reasoning?

What is fundamentally wrong in writing $(-a)^{1/2}$ as $((-a)^{2})^{1/4}$ when $a$ is positive and thus equating it to $a^{1/2}$? Edit: I'm basically asking if there is anything wrong with this operation like multiplying $1$ and $2$ with $0$ and…
Vulgar Mechanick
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What is the meaning of exponentiation?

For natural numbers $a,b$ the operation $a^b$ is defined as $a * a * a * a \cdots b$ times. However when you get to negative powers, complex powers, complex bases, polynomial expressions etc then this definition falls flat. So what exactly is…
Physicist
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Solving $\sqrt{x}^{\sqrt{x}^{\sqrt{x}}} = 2^{512}$.

I can't solve this question I have tried but I can't find any other websites that help. Thanks $$\sqrt{x}^{\sqrt{x}^{\sqrt{x}}} = 2^{512}$$ This is not infinite exponent just 2 times
Unknown
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definition of exponents

if you wanted to define $a^x$ for all integers would this be ok: For non negative integers: $a^0=1$ $a^{(n+1)}=a^n\cdot a$ For negative integers: $a^{-n} = \frac{1}{a^n}$ an identity which holds for $n$ a positive integer, by definition. So what…
James Doherty
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Solve $2(4^x) + 3(9^x)=5(6^x)$

Ended up with $2^{2x+1} + 3^{2x+1} = 5(6^x)$. Not sure how to proceed from here. Thanks!
user445310
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So what's the difference between $(x^a)^b$ and $x^{a^b}$?

So what's the difference between $(x^a)^b$ and $x^{a^b}$? Particularly, why does Wolfram Alpha treat them differently? http://www.wolframalpha.com/input/?i=(x%5E3)%5E2 http://www.wolframalpha.com/input/?i=x%5E3%5E2
mavavilj
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Negative exponents problem.

I am very confused on this question. How does $(4^{-3} \cdot 2^{-3})^0 = (8^{-3})^0$? Should it not be $8^{-6}$ since $-3+-3 = -6$? I just don't get where the other power of $-3$ goes off to?
Mobius-User
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What is the units digit of the product of several numbers

This problem is quite challenging to me. I know the answer should be an even number, but not know how to solve the problem. Thank you for help! What is the units digit of the product $2^1·2^2· 2^3 · 2^4 · · · 2^{199}· 2^{200}$ where the numbers…
user321527
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finding the second smallest number

Please tell me how to solve this kind of problem in a fast manner. Which of the following is the second smallest number: $2^{120}$, $3^{80}$ and $10^{30}$? Hope you could show me the best solution, thanks.
Stephen Mark
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Solve Equation for n where n is the power ($3^n = \frac{1}{81}$)

I have the equation: $$3^n = \frac{1}{81}$$ And I need to find n. Can someone explain how I do this, with steps please (GCSE level)? TIA.
Matt Cowley
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