# Questions tagged [exponentiation]

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

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### How to construct exponentiation over all of $\mathbb{R}$?

I want to construct exponentiation over $\mathbb{R}$. $$^\wedge:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$$ Using a strictly constructive approach $\mathbb{R}=\{[(x_n)]|(x_n)\in\mathbb{Q}^\mathbb{N}\}$, ...
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### Trying to prove $(a^x)^y=a^{xy}$ using definition

In a lecture, we learned that $a^x$ is defined to be the limit of $a^{x_n}$ where $x_n$ is a series of rationals that converge to x. I was trying to prove $(a^x)^y=a^{xy}$ using solely this definition,...
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### Prove/disprove that $a^{2m} + b^{2m} + c^{2m} > 2^{1-m}$ subject to $a + b + c= 0$ and $a^2 + b^2 + c^2 = 1$

Problem. Let $a, b, c$ be reals with $abc\ne 0$, $a + b + c = 0$, and $a^2 + b^2 + c^2 = 1$. Prove or disprove that $a^{2m} + b^{2m} + c^{2m} > 2^{1-m}, \forall m\in \mathbb{Z}_{>2}$. Prior ...
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### The value of $\sqrt{(-1)^2}$? [duplicate]

Which is the correct value of $\sqrt{(-1)^2}$ ? $\sqrt{(-1)^2} = \sqrt{(-1)\cdot(-1)} = \sqrt{1} = 1$ $\sqrt{(-1)^2} = ((-1)^2)^{1/2} = (-1)^{2\cdot1/2} = (-1)^1 = -1$ I was surprised to realize ...
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### second power in exponential

I am struggling with raising some powered expression to another power, like $(a^b)^c$ which should equal to $a^{bc}$. As two examples $(3^2)^3=9^3=729=3^6$ $3=3^{2/2}=(3^2)^{1/2}=(3^{1/2})^2$. Now if ...
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### What are the x intercepts (wherever defined) for $\vert x^2 -8x +15 \vert ^ {(x-1)(x-2)(x-3)(x-5) \over (x-2)} = 1$

The original question asked was the following : (Source : https://drive.google.com/file/d/1U7khRKh12GlaSY73210oQd2170JiXEGY/view) Rita took some of her friends for picnic. Her friends are at x−...
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### How to estimate $10^{\frac{1}{2}} + 10^{\frac{1}{3}} + \ldots + 10^{\frac{1}{10}}$

I heard an interesting interview question recently, which was as follows: Estimate the value of $X = 10^{\frac{1}{2}} + 10^{\frac{1}{3}} + \ldots + 10^{\frac{1}{10}}$. You have 30 seconds to Compute ...
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### Generalization of the matrix exponential [closed]

I've seen this post which addresses the question of exponentiating a vector. I was wondering if there's a well-defined notion of exponentiating a rank $r$ tensor? For instance, if I have a rank 3 ...
### Why does $x^{\frac{3}{4}}$ have a domain of $x≥0$, while $x^{\frac{6}{8}}$ has a domain of $x\in{\Bbb{R}}$? [duplicate]
Why does the simplified version of $x^{\frac{6}{8}}$ have a domain of $x≥0$ while the unsimplified has a domain of $x\in{\Bbb{R}}$? Shouldn't they have the same domain being that they are the same ...
I can't seem to demonstrate how $\frac{\sqrt{\sqrt{3}+2}}{2}$ is equal to $\frac{\sqrt{2}+\sqrt{6}}{4}$... even though my calculator tells me that both are equal $0.965925826$. Hoping someone can show ...