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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Need help in solving an exponential equation

Find integer $x$ so that $$5\cdot5^{2x}-24\cdot2^x - 5 = 0$$ I know there is an approach as we can show the function $f(x):= 5\cdot5^{2x}-24\cdot2^x - 5$ is injective. So, the equation can have at most one solution. And, basically, if we can guess…
Akhtubir
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Given $a,b,c$ solve $1=x^a + x^b - x^c$ for $x$

If $a,b,c$ are constants, is it possible to isolate $x$ in the following equation? $$1 = x^a + x^b - x^c$$ If so, how do I achieve this?
fuseinabowl
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Operation signs as exponents

I sometimes see numbers or constants or variables having exponents the operation signs, such as these: $a^{+},~ a^{-},~ (1/4)^{+},~ k^{\pm},~ 0^{+},~ 0^{-}$ All I know is the zero to the power of plus and minus, which happens for me to know it from…
serkan
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Why isn't $x^\frac{-1}{2}$ equal to $-\sqrt{x}$

A fraction multiplied by $-1$ can be written in different ways: $\frac{-a}{b}$ $\frac{a}{-b}$ $-\frac{a}{b}$ So $x^\frac{-1}{2}$ can also be written as $x^\frac{1}{-2}$ then why can't we take the whole $-2$ down and turn it into $-\sqrt{\bullet}$…
Manar
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How this is true for laws of exponents?

If R = $(\frac{V}{pi})^\frac{1}{3}$ If I ask you to find $R^2$. We substitute it and make the power 2/3 as I saw in my textbook. Why didn’t the powers add if they are multiplied.
user848010
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Taking exponents of two numbers to make them equal

If $3^a = 5^b = m$ and $\frac{1}{a} + \frac{1}{b} = 2$, what is the value for m? Since $a, b \neq 0$, I assumed both a and b are fractions, so I did: Let $a = \frac{a_1}{a_2}; b = \frac{b_1}{b_2}$ Then we subsitute them in and get $3^{a_1 b_2} =…
Cyh1368
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$(|x| + \sqrt {x ^ 2 - 1}) ^ x + (|x| - \sqrt {x ^ 2 - 1}) ^ x = 2(2x^2 - 1)$

I have come across the equation $$(|x| + \sqrt {x ^ 2 - 1}) ^ x + (|x| - \sqrt {x ^ 2 - 1}) ^ x = 2(2x^2 - 1)$$. Of course we have $x \in (-\infty, -1) \cup(1, \infty)$. I noticed that $(|x| - \sqrt {x ^ 2 - 1}) ^ x = \frac{1} {(|x| + \sqrt {x ^ 2 -…
andu eu
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What is an example of zero exponent in nature?

This concept is difficult for non-professionals to grasp, and I admit that I can't even conceive of how this exists in nature, as opposed to proving that 2+2=4 by a more traditional explanation such as grouping objects together. Does some number…
WannabeHolyPerson
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How to solve for $x$ in the equation $y = (x + 25)^x$ where you know the value of $y$

I am trying to figure out how to solve for $x$ in the equation $y = (x + 25)^x$ for any known $y$. For example, I know that in the example $2758547353515625 = (x + 25)^x$, the value of $x$ is 10. Is there a way to solve for $x$ for any known $y$?
yDivide
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How to get the value of r when I have the value of r^² ( 1000 = r^2)?

I do not remember this from school anymore and I now I have to use the logarithm but the problem is that I do not know the base. I want to know the value of r but I only have the value of r^2. 2² = 2*2 and the inverse is log2(4) but in my case how…
Kaspacainoombro
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Why is $\sqrt{x^2}$ is $|x|$?

I was trying a problem and was getting the wrong answer and when I saw the solution on the internet I found this statement written in square brackets $\sqrt{x^2}$[note square is on $x$] is $|x|$. Till now I have learned that by laws of exponents we…
Shreyansh Kuntal
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I don't understand this exponent simplification

I've been doing the Khan Academy math courses to brush up on my math foundations before starting my CS/math degree in the fall semester. I just don't fully understand negative exponents, I stumbled upon the following exponent…
Imperator
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how to define $x^\sqrt 2$?

Can anyone please tell me how to define $x^\sqrt 2$ ? If it was $x^2$ or $x ^ \frac{1}{2}$, we could have said that $x^2$ means $x \times x$ and $x ^ \frac{1}{2}$ means a number y such that $y^2 = x$. But how to define $x^\sqrt 2$ ? Can anyone…
anonymous
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Can decay percent be less than $-100\%$?

I have this problem: $$A=A_0\left(1+\frac{r}{n}\right)^{nt}$$ where $A=100$, $A_0=25$, $n=1$, and $t=2$. This leads to \begin{align*} (1+r)^2=4 &\quad\Rightarrow\quad |1+r|=2\\ &\quad\Rightarrow\quad 1+r = +2 \text{ or } -2 \\ &\quad\Rightarrow\quad…
ahmed allam
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exponent rule when dividing the same exponents

I ran into a bit of confusion when applying the exponent rule: $x^a/x^b = x^{a-b}$ Then when $4^x/2^x$ why does it equal $2^x$ if we apply the rule above then wouldn't it be: $(4/2)^{x-x}$ ? Thank you!
Katie Melosto
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