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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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Largest number below a bound expressible as the exponentiation of two natural numbers

What is the largest number less than a given $N$ expressible as $b^e$, where $b,e\in\mathbb N$? For $N=10$ the answer is $3^2$; for $N=18$ the answer is $2^4$. Is there a procedure to find the correct $b$ and $e$? Here $b$,$e$>1 Note: Only $N$ is…
Hemachandra
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How solve for x in an Infinite exponent

How would one solve for x in the following equation: $x^{x^{x^{x^{\cdots}}}} = 4$ The exponent continues forever... So what is the value of x? Thank you for helping
Demetrio MacGuire
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Is an equation of the following form solvable?

Is it possible to solve for $x$ which satisfies the equation $$d=(a\exp(bx)+\exp(cx))x^2$$ where $a,b,c,d$ are given constants? It looks quite horrible... Many thanks!
emory j
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How to solve $1/{4a^{-2}}$

Can i write $1/4a^{-2}$ as $4a^2$ ? Or is the right answer to do it like: $$1/4a^{-2} = 1/4 \cdot 1/a^{-2} = 1/4 \cdot a^2 = a^2/4$$ In the problem there is no parenthesis around $4a$ but assuming there were parenthesis like $1/(4a)^{-2}$ would it…
user346461
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How do I get negative exponent?

I am trying to get negative exponent expression to work. Say I have 1.3 * 10^-3 I tried putting that between \$$ pair and got the following $$1.3 * 10^-3$$ I see 10-3 instead of seeing -3 in superscript Thanks
fahadash
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Express $(100^3)^5$ with a base of $10$

Express $(100^3)^5$ with a base of $10$. I don't get this.
Anonymous
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Geometric definition of logarithm (finite and non-kinematic)

Is there a geometric definition of the logarithm function that is non-kinematic and does not involve an infinite procedure? With base 10, for example. I'm asking for definition, not construction method. (Napier's 1619 definition was kinematic in…
exp8j
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simple question, need help

Can someone tell me where does 1 come from on the end, this got me really confused.
Dovy
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Writing a non integer power in terms of integer powers

I would like to write $x^{2.5}$ in terms of $x$ to the power of integers, is there any way to do this. Taylor series etc. don't work when they depend on derivatives. If it is not possible, do you have or know a proof. Thanks EDIT: to clarify, I mean…
John Echo
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Subtraction of large numbers with exponents

Is there an easy way to break down the below formula? And make it easy to calculate it mentally without the use of a calculator? $108^2 - 92^2$ I know this is probably very basic, but I cant remember how.
matolv
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Expression evaluation

I have encountered this expression and I cannot evaluate to the desired result on the right side $$3\cdot4^{k+1}+4^{k+1}-64 = 4^{k+2}-64$$
john smith
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How to calculate $\lim \limits_{h \to 0}{\frac{a^h-1}{h}}$?

As the title says, I would like to prove for $f(x) = a^x$ there is always some constant c such that $f'=cf$. Is calculating the limit the right approach to solve this problem? Also, how to show there is only one solution when $c=1$? (the $e^x$)
Isabel Yang
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-1 raised to fractional indices lying between 0 and 1

For a personal project, I had to figure out what happens when $-1$ (negative one) is raised to fractional powers lying between $0$ and $1$. I thought that if I get a power $x = 0.a_1a_2a_3...a_n$ (where $a_1,...,a_n$ are digits), then I could…
Nilabro Saha
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How many times is $3^{20} + 3^{22}$ greater than $3^{20}$

I just don't know where to start any help is appreciated.
Rayyan Merchant
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Evaluation of $x^{y^{z}}$

Whether $x^{y^{z}}$ should be considered as $x^{\left ( y^{z} \right )}$ or $\left ( x^{y} \right )^{z}$, without any context? If any one among these two is default consideration? $\left ( x^{y} \right )^{z}$ appears more natural to me, but my…
Romy
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