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 wrong in my calculation ($x^{-1} \cdot \sqrt[3]{x} = ?$)?

I need to calculate the following product: $x^{-1} \cdot \sqrt[3]{x} = ?$ First, I apply the rule $\sqrt[n]{x^m} = x^{\frac{m}{n}}$ to convert $\sqrt[3]{x}$ to $x^{\frac{1}{3}}$: $x^{-1} \cdot \sqrt[3]{x} = x^{-1} \cdot x^{\frac{1}{3}}$ Then I add…
Glory to Russia
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$x^{5x}=y^y$, $x, y \in \mathbb{Z}^+$, find largest value of $x$.

Let $x$ and $y$ be positive integers satisfying $x^{5x} = y^y$. What is the largest possible value for $x$? I'm stuck on this question in an Olympiad past paper. Anyone have any ideas about this one?
Maths explorer
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Exponent notation (Tetration)

What is the meaning of this notation, and how does it work? $$\exp_{10}^2(1.09902),\,\exp_{10}^3(1.09902)$$ I knew this notation when i was reading about tetration on wikipedia. Here is the link: Tetration and the values ​​above are equal to the…
user516076
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Behaviour of Exponents

Something that many early students, including myself, take for granted is that $$x^\frac{3}{2}=\sqrt{x^3}=(\sqrt{x})^3$$ but is this true? Is exponentiation "commutative" and does a fractional exponent mean the same thing as a root?
Colin Hicks
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When do rational exponents produce two answers?

I am wondering when rational exponents produce two answers. I am aware that X^2=25 produces two answers and why this is the case. I have not found a good resource for every case. What is (-2)^(2/2)? If we take the square root first, do we need to…
yourmoveantonius
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Find two numbers $a,b$ such that $a^a=b^b$

Playing around on Desmos with the equation $y=x^x$, I noticed that the function, for numbers $0
DonielF
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Compare numbers with big powers

Compare $2019^{2020}$ and $2020^{2019}$. I know that $2019^{2020}$ is greater than $2020^{2019}$ but I couldn't prove it. I tried proving $(\frac{2019}{2020})^{2019}.2019>1$ but without success.
mathlover
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Negative exponents

By a coincidence I have found that $10^{-6,7} \approx 10^{-7} \times 2 $. This just seems extremely random to me, why is this so? First I thought it might have to do with the fact that $6,7 \approx 10\times \dfrac{2}{3}$ but I don't know if that is…
Serendipity
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Exponential equation, $(3+2\sqrt2)^x+1=6(\sqrt2+1)^x$

https://i.stack.imgur.com/YP2Ha.png $$(3+2\sqrt2)^x+1=6(\sqrt2+1)^x \qquad\qquad x\in\mathbb{R}$$ I managed to find one of the solutions (x=2), but I got stuck. I would really appreciate a step by step solution. Thanks in advance :)
Andreea
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How to calculate $n$ in $f = p^n - q^n$?

I have the formula: $f(n) = \frac{p^n - q^n}{\sqrt 5}$ Assuming I know the value of $f(n)$, can I calculate $n$? Sure, I can convert to $ \sqrt 5*f(n) = p^n - q^n$ . But after that I stuck... edit 1 $p = \frac{1 + \sqrt 5}{2}$ $q = \frac{1 - \sqrt…
Superluminal
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Smooth transition between functions

Therefor that I don't really have a mathematical background, it is kind of difficult to me, to describe what I'm looking for (but I'll give it a try): I'm looking for a way to parameterize a function to fulfill the following constraints: function…
Tom Mekken
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Does the fact that $x^2=(x-1)(x+1)+1$ have a name?

Just curious about this pattern $$x^2 = (x-1)(x+1) +1$$ So: $$\begin{align} 1^2 &= \phantom{1}0\cdot\phantom{1}2+1 = 1 \\ 2^2 &= \phantom{1}1\cdot\phantom{1}3+1 = 4 \\ 3^2 &= \phantom{1}2\cdot\phantom{1}4+1 = 9 \\ 4^2 &=…
stramin
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What is wrong with this problem

We know that: $(a^n)^m=a^{nm}$ From this we have: $-3^3=[(-3)^2]^\frac{3}{2}=(3^2)^\frac{3}{2}=27$ Find what's wrong
user123451241
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Exponent Law: Negative Exponents Division

I'm trying to relearn high-school maths after years of decay. This is a very basic exponent law question. How can I prove the following in a step-by-step fashion: $$ \biggl(\frac{7^3}{3^8}\biggl)^{-2} = \frac{3^{16}}{7^6} $$ Thanks!
Zilong Li
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How is the definition for exponentiation extended to rationals and reals?

If you have defined multiplication, division, addition and minus on all the real numbers, then you can define positive exponents by $b^1=b$ and $b^{n+1}=b^n\cdot b$. From this definition it is possible to prove the following identities for positive…
James Doherty
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