Questions tagged [exponentiation]

Questions about exponentiation, the operation of raising a base $b$ to an exponent $a$ to give $b^a$.

3,063 questions
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How to convert any base 10 integer to a rational power of 2?

For example, through (lots of) trial and error with a calculator, I figured out that: 1.551e+25 is approximately equal to 2^83.6814 Is there an equation or algorithm for a conversion like this? ...
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Hard exponential equation

Is there any algebraic way of solving the following equation for $x$? $$\frac{3^x+2^x}{3^x-2^x}=7$$ Apparently there is some way of solving this and I heve tried to solve it in a conventional ...
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Is it true that $n \leq 2^{n-1}$ for all natural numbers $n > 0$? [closed]

Seems to be true but I want to make sure: $n \leq 2^{n-1}, n > 0$
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How to decelerate from velocity v to stop time t over distance d?

I'd be grateful for some help with this problem I am trying to solve. Let's say that I have an object travelling at a velocity v. I want that object to come to a halt in time t AND travel exactly ...
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What is the difference between $\dfrac{1}{3^{-2}}$ and $3^{-2}$

Stupid question but how in the world does 1/3^-2 not be the same thing as 3^-2? The first answer I got is 9, the second one I got is 0.111... Isn't the negative power just 1/3/3? What difference does ...
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Why do we take $(1+z)^{\alpha}$ as $e^{\alpha \operatorname{Log}(1+z)}$?

Let $\alpha$ be a complex number. Show that if $(1+z)^{\alpha}$ is taken as $e^{\alpha \operatorname{Log}(1+z)}$, then for $|z|< 1$ \begin{equation*} (1+z)^{\alpha} = 1 + \frac{\alpha}{1}z + \...
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Near integers in powers of binomials with radicals

This question comes out of a mathematics calendar problem that asked for the tenths digit of the expression $(17 + \sqrt{280})^{17}$. The calendar implied the digit should be 9, but after playing ...
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How to show this rule works for whole numbers to the fifth power?

Today I was shown a rule about natural numbers raised to the fifth power and an interesting method to generate them through the odd numbers. Start with $1 = 1^5$. Then skip the next $T_1 = 1$ odd ...
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How do I express $y = 1 - e^{-ax^b}$ as a linear function in terms of $y$, $x$, $a$, and $b$?

I tried using the log rules and eventually got to $\ln(\ln(1-y)) = \ln(-a) + b\ln(x)$, but $\ln(-a)$ is obviously not defined across all $a$. This assumes that $x\ge 0$ and that $y<1$.
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Cube pattern of last few digits [duplicate]

Consider the sequence $$t_0 = 3 \qquad t_1 = 3^3 \qquad t_2 = 3^{3^3} \qquad t_3 = 3^{3^{3^3}}$$ defined by $$t_0 = 3 \qquad \text{and} \qquad t_{n+1} = 3^{t_n}, \qquad n >=0$$ What are ...
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Determine all real (a,b,c) that satisfy the systems of equations.

Determine all real (a,b,c) that satisfy the equations $a^2b^3c=-(2^7)$ $ab^2c^3=(2^8)$ $a^3bc^2=-(2^6)$ I have tried making every variable a base 2 (and (-2)) to an exponent and then use systems ...
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Properties and applications of complex logarithms and exponentials.

We have these two identities where $a>0$ real, $n$ integer and $z$ complex : \begin{align} (a^z)^n=a^{nz} \tag{1} \\ (a^n)^z=a^{nz} \tag{2} \end{align} $(1)$ is true. Is $(2)$ also true? ...
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Addition of different bases raised to the same power (solve for power). Is power determined uniquely?

Let a vector $\mathbf{x} = [x_1, \ x_2, \ \dots \ , \ x_n]' \in \mathbb{R}^n$ contain non-negative elements i.e. zeros and positive values. Problem: $x_1^a + x_2^a + \dots + x_n^a = c \quad$ ...
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What happens when a is negative in $a^{\frac{m}{n}}$.

What happens when a is negative in $a^{\frac{m}{n}}$. This question may seem silly but I got confused by it. If you take the example $2^{\frac{4}{3}}$ Method 1: $\sqrt{2^4} \approx 2.51984$ ...
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