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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Is there a way to simplify this exponent?

I have the exponents $k^4+k^5+k^6$. Is there any way to simplify this into one exponent? I'm trying to find a way to simplify a sequence of increasing exponents into one, but am not sure how. With some guess and checking, I got $k^4+k^5+k^6 \approx…
Blackhole927
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Find x and y for $11^x=5*2^y+1$

PROBLEM Solve the equation in the set of natural numbers: $11^x=5*2^y+1$ WHAT I THOUGHT OF We can write $11^x$ as $(10+1)^x$ We know that $(a+b)^x= M_a+b^x=M_b+a^x$ Applying the formula above we can write $(10+1)^x=M_1+10^x$ $M_1$ basicly means…
IONELA BUCIU
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Is there any way express $2^{6/x} - 2^{x+1} + 12 = 0$ taking $2^x$ as some variable, say $a$?

I tried raising to $x$ on both sides, getting $2^6 - 2^{x(x+1)} + 12^x = 0$, but we still can't simplify the $2^{x(x+1)}$ in terms of $2^x$. Is there any method of simplifying the equation, taking $2^x$ as some variable, say $a$? Edit: I got this…
Bongo Man
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Do I understand negative exponents correctly?

I know that each time you go up an exponent, you multiply by the base again e.g. $2^2$ to $2^3$. To go down, you you divide by the base. The way I find negative exponents is that I divide 1 by the base^exponent E.g. $10^{-3} =$ ? $10^0 = 1$ $1 ÷ 10…
Haseen Siddiqui
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Raising something to the zeroth power is one. This works in many contexts. What is the connection between the contexts?

I learned that the reason $a^0$ is defined as $1$ is for consistency with the rule $a^ba^c=a^{b+c}$. However, the fact that $a^0$ is $1$ is used in contexts outside of the algebraic manipulation seen in the prior sentence. It is unclear to me what…
Dargscisyhp
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Power law when x = 0, but actual y is greater than zero

I am trying to fit various equations to my data. It looks like the derivative of a power law model ($y=ax^b$) will work pretty well. So an equation of the form: $$y=abx^{(b-1)}$$ However, I am unclear as to what happens when $x = 0$. From the…
rdemyan
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Decreasing power?

Say you have numbers X and Y When Y is 1 you want to times X by .5, When Y is 2 you want to times X by .5 then by .25, When Y is 3 you want to times X by .5, then by .25 then by .125, and so on. How can I do that in a spreadsheet? It's like a ^ but…
skybreaker
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Why is $2^{2^2}$ so much less than $2^{2^{2^2}}$?

I assume there's a name for raising something by an exponent repeatedly, but I haven't been able to find it. I understand why $2^{2^2} = 16$ and $2^{2^{2^2}} = 65536$ by plugging in the numbers, but I am having trouble building some intuition for…
Each One Chew
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Why do we need a positive base for a fractional exponent

In my school book it says that if $f(x) =$ $x^{\frac{m}{n}}$. The base ($x$) should be greater than $0$ or in other words $Af=(0,+\infty)$. Why is that? If the base is negative I run to the contradiction that while $(-1)^3 = -1$, if you write it as…
Xristos Ntallas
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Indices rule for division with 0

The indices rule for division states $ m^x \div m^y = m^{x-y}. $ This works out because $ m^4 \div m^2 = {{m*m*m*m} \over {m*m}} = m*m = m^2 = m^{4-2} $ So we see that through the reduction rule for division, $m^y$ cancels out. In my mind, this…
The Fool
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How to calculate $x,n$ for $x^n=400$

Basically, what I'm trying to figure out is: If I have a directory tree, and each level of the directory tree has $x$ directories, and there are $n$ levels, then the number of directories at level $n$ is $x^n$. So, If I want to generate a structure,…
Leshawn Rice
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What are the natural solutions $x, c$ to $2^x + 1 = c\cdot 3^2$ for odd $x$?

I understand that $x=3, c=1$ is a solution. Are there any more natural number solutions? If so, how would I find them? If not, why not?
IAAW
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Multiplication/Division of Exponents

I have the following two equations: \begin{align} \frac{\lambda}{u_L} = 3.2693\left(\frac{\nu}{\epsilon}\right)^{1/2 }\end{align} \begin{align} u_L = \left(\epsilon \lambda\right)^{1/3}\end{align} I have tried to solve for $\lambda$ as…
rdemyan
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Given $x=a^b$ and $y=b^a$, find solutions for $a$ and $b$ in terms of $x$ and $y$.

My first approach involved simply taking the product: $$xy=a^bb^a=b^{b\log_b(a)+a}$$ However, I can't really do anything to, for instance, separate the exponent from the base $b$. Additionally, division has basically the same problem, and addition…
Zuter_242
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Exponential difficulty

I am finding it difficult to do the $(2(2^{-x})$ part of the problem below. So far I have gotten $2^{x+1}$ to equal $2y$ and I cannot seem to find how to get $2(2^{-x})$ $2^{x+1} + 2(2^{-x}) -5 = 0$ $2^x = y$ solve for x
Highlandword9
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