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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Solving equation where unknown is on the exponent

In general is there a systematic of solving equations of the following form: $$ a \exp(bx^2+cx)+ d \exp(x) = e $$ where $a$, $b$, $c$, $d$, $e$ are constants and $x$ is the unknown.
Sam
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Stuck on Elementary Exponentiation

I'm confused. Why is it that for a problem in the form of: $(2^{x+1})(2^{x-1})$ we get $2^{2x}$ instead of $4^{2x}$; Shouldn't we multiply the $2$s by each other..? Similarly for a problem like: $(2^{x+1})(4^{x-1})$, why do we get $2^{3x-1}$ rather…
jaykirby
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Multiplication of variables to power

I got this equation $$MRS= - \frac{3/4x^{1/4} \ y^{-1/2}}{\frac{}{}3/2x^{-1/4}\ y^{1/2}}$$ As you can see the absolute values of the exponents are the same so they should cancel out each other somehow, but I don't know the rule. Can somebody explain…
Ivan Hanák
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How do negative powers and fractional powers make sense?

We know that powers are the number of times a number is multiplied with itself. Like $2^3$ means $(2×2×2)$, $5^4$ means $(5×5×5×5)$, etc. But how do negative powers make sense? What does it mean to multiply a number negative number of times by…
AltercatingCurrent
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Can anyone help me understand what this equation is?

Is there a way to express this equation where you have an integer $x$ and an integer $n$ and you do: $$ x^n + x^{n-1} + x^{n-2} + ... + x^{n-n} $$ Is there a name for this type of equation and how can I write it more succinctly? I haven't done math…
user3006927
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The result of a negative number raised to equivalent fractional powers

When a negative number is raised to a fractional power, will two equivalent fractions as powers give the result the same sign (i.e., both negative or both positive)? For example, I want to raise -5 to a power of $\frac{1}{3}$. $$(-5)^\frac{1}{3}$$…
Cynicrom
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raising to the same power

when we add the same number many times we can use multiplication for shorter: $$2+2+2+2 = 2\times4$$ when we multiply the same number many times we can use exponentiation for shorter: $$2\times2\times2\times2 = 2^4$$ is there next level of shorter…
Tomek Wyderka
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Expressing exponential form to trigonometric form

Express $e^{-\frac{1}{2}i\theta} - e^{\frac{1}{2}i\theta}$ in trigonometric form, and show that $(1-e^{i\theta})^2 = -4e^{i\theta}\sin^2\left(\frac{1}{2}\theta\right)$ By inspection, I noticed that the former expression is identical to…
lulu
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For every positive integer $a$, find a composite number $n$ such that $n|a^n -a$.

For every positive integer $a$, find a composite number $n$ such that $n|a^n -a$. Solution: We have $ a^n - a$ $ = a[a^{(n-1)}-1]$ If a=n, then $n|a^n -a$ Is this the way to do it or do I need to do it in another way?
Anwesa Roy
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What is the most effective way to calculate the exponent in $x^y = z$?

I was recently watching a video on Karatsuba's fast multiplication algorithm and the narrator stated something that intrigued me: $2^{1.6} \approx 3 $ Specifically, I wondered what power $2$ would need to be raised to in order to be equal to…
Hazel へいぜる
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How to visualize exponents

One thing that always confuses me about exponent is that how to visualize it. Suppose we have $$2^3$$ This means that we multiply 2 three times $2^3 = 2\times2\times2$ But what if the exponent is a decimal such as $2^{2.345}$ Does this mean that we…
Kain
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Is $\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$ true for all complex numbers?

I have just started learning complex numbers and am now confused about one property. We know that $\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$ is only valid if both $a$ and $b$ are not negative simultaneously. My teacher told me that the relationship breaks…
Habib
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Is $1^{1/0}= 1?$

If: $1^0 = 1$ then: $1 = \sqrt[0] 1$ but that implies that: $1 = 1^{1/0}$ But since 1/0 doesn't exists, how is this possible?
pollatron
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Is there a shorthand function for this repetitive equation

I was looking for a way to get the answer to this equation without having to repeatedly type out each number $$ x^{y}+x^{y+1}+x^{y+2}+...+x^{y+n} $$ What is the smallest equation that outputs the same answer as this one?
Vexian Empire
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Why is (virtually) anything raised to $0$th power equal to $1$?

Taking a shot in the dark: $10^3 = 10 * 10 * 10 = 1 (10 * 10 * 10)$ $10^2 = 1 (10 * 10)$ $10^1 = 1 (10)$ $10^0 = 1$ This isn't quite proofy enough, but seems to be reasonably convincing to a non-mathematician like myself. Is there a deeper reason…
jbuddy_13
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