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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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If $a^{(b^c)}=d^c$, find $d$ in terms of $a$ and $b$.

Is it possible to express $d$ in terms of $a$ and $b$ only in the following equation? $$a^{b^c}=a^{(b^c)}=d^c$$ I want something like $d=\dots$ Thanks in advance!
GamrCorps
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Find the number of times we can take log base two of this exponent

I'm not sure how to simplify this type of exponential expression. I would like to know $k$ many $log_{2}^{k}(n)$ such that $n \leq 1$ $$n = (2^{2^{2^{15}}})^2$$ So attempt I to simplify $$n = 2^{(2^{2^{15}} ) + (2^{2^{15}})}$$ From here I'm not…
lzc
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How to simplify algebraic expression with exponents?

Can someone please tell me step-by-step how to simply this? $$\sqrt{\frac{8x^{{\frac{1}{2}}^{\frac{2}{3}}}}{x^{-\frac{1}{2}}}}$$ Edit: correct answer is 2x^(5/12), I'm just not sure how to do it. Thank you!
Spica
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Negative number squared in expression $-5-8^{2}$

I know this probably is a silly stupid question, but I just don't get it. I'm currently doing Khan Academy pre-algebra and stumbled upon an awkward problem. I assume that: $-5-8^{2}=59$ Because -8 squared is 64 and et cetera. But the right answer…
wswld
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Definition of Zeroth Power

What is the definition of raising a number to the zeroth power ($x^0$)? I know that many people say that "anything raised to the zeroth power is one" but this is clearly not true since $0^0$ is $undefined$. How then do mathematicians define $x^0$…
terminex9
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How to formulate exponential growth?

Here's my question: A rumour spreads exponentially through a college. 100 people have heard it by noon, 200 by 1pm. How many people have heard it a) by 3pm b) 12.30pm c)1.45pm thanks in advance.
Deniz
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Why is $(-1)^x=e^{i\pi x}$

I was recently taught exponentials and I decided to play around with negative bases, which they told me were not allowed. The obvious place to start was negative one, and, as expected, the graphing tool did not work. However, after trying Wolfram,…
GuPe
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Is $a^b$ larger than $b^a$ if $a 1$?

Is $a^b$ larger than $b^a$ if $a 1$? I tried this out for a few numbers and this seems to be the case. If this is true, could you show me a proof? I would be very interested. If this is not true, can you give me a counter example? I…
Héctor van den Boorn
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Trailing zeros in indice question

The expression $15^{80}$ x $28^{60}$ x $55^{70}$ gives a number that ends in a string of zeros. How many consecutive zeros are in that final string? I've done this type of question with factorials, but I've no idea how to approach this with indices.…
numbermaniac
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Need assistance solving exponential equation: $64=0.8^d$x$100$

Solve the exponential equation: $64=0.8^d \cdot100$ I tried doing: $64/100=80/100^d$ but since there is no common factor which gives these numbers with different powers I failed to find the value of variable $d$. How to solve it then?
Arthur Alex Karapetov
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Harmonic Mean Solution

The harmonic mean of two positive numbers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean equal to $6^{20}$? I don't really know how to go…
user251865
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What is the value of i^i?

i is an imaginary number. What is $i^i$? I tried to use euler rule but the answer is strange. For example $i = e^{\frac{1}{2}i\pi}$. Using $(a^b)^c = a^{b*c}$ we got $i^i=e^{(\frac{1}{2}i\pi)*i} = e ^ {-\frac{1}{2}\pi}$ Doesn't seem right The…
user4234
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Values of $a$ for which the equation $100^{-\lvert x \rvert} - x^2 = a^2$ has the maximum amount of solutions

$100^{-\lvert x \rvert} - x^2 = a^2$ I don't know how to approach this problem, due to the x in the exponent. I would appreciate hints more than outright solutions :)
John Doe
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exponent problem solving

I came across a problem; $a^x=b^y=c^z$ and $b^2=ac$. It is required to show $\frac{1}{x}+\frac{1}{z}=\frac{2}{y}$. I have tried the following steps- \begin{equation*} b^2=ac \\ b=\sqrt{ac} \\ b=a^{1/2}\ast c^{1/2} \end{equation*} So…
Soham
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Why is 'something per hectare' denoted with a negative exponent ( $ha^{-1}$)?

Quick question.... why is it that something per hectare is shown as having a negative exponent, $ha^{-1}$? For example, on this page: http://www.ipcc.ch/ipccreports/sres/land_use/index.php?idp=12 1 tonne per hectare is shown as (t $ha^{-1}$). I did…
AggroCrag
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