# Tagged Questions

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

2answers
37 views

### Powerseries of $b^x$

I read on wikipedia that the exponential function $e^x$ can be written(defined) in this form $\displaystyle e^x:=\sum_{n=0}^{\infty}({1\over n!})x^n$ So my question was if its is then possible to ...
2answers
32 views

3answers
85 views

### If $n$ is an odd integer, does $x+y$ divide $x^n + y^n$?

I believe that the answer is yes. Here's my thinking: $x^n + y^n -(x+y)(x^{n-1} + y^{n-1}) = -x^{n-1}y -xy^{n-1}$ $-x^{n-1}y -xy^{n-1} - (x+y)(-x^{n-2}y -xy^{n-2}) = x^{n-2}y^2 + x^2y^{n-2}$ So, at ...
1answer
171 views

### Is The Statement $b^n\equiv 1\pmod n$ equivalent to “$x\mapsto b^x-x\pmod n$ is a bijection”?

Suppose that $n$ is a natural number and $b$ is one coprime to it such that $b^n\equiv 1\pmod n$. Does it follow that, if $b^x-x\equiv b^y-y\pmod n$, then $x\equiv y\pmod n$? This is inspired by the ...
4answers
70 views

### Why $2(0.3)^2$ doesn't equal $0.6^2$

Why $2(0.3)^2$ doesn't equal $0.6^2$? I mean if $0.6 = 2(0.3)$, then why $2(0.3)^2$ doesn't equal $0.6^2$? I think it is because of the power but I'm not sure about that. All that I know is that it ...
4answers
872 views

### What is the *middle* digit of $3^{100000}$?

The decimal representation of $3^{100000}$ has $47713$ digits. What is the $23857^{th}$ digit - i.e. the one in the $10^{23856}$'s place? There are lots of questions on this site asking for the last ...
6answers
344 views

### Finding the mod of a difference of large powers

I am trying to find if $$4^{1536} - 9^{4824}$$ is divisible by 35. I tried to show that it is not by finding that neither power is divisible by 35 but that doesn't entirely help me. I just know that ...
6answers
208 views

### Why is $\ln(x^x)=x\ln(x)$ valid?

I know that $\ln(x^k)=k\ln(x)$ for any constant $k$, but why is $\ln(x^x)=x\ln(x)$. The exponent $x$ is not constant.
6answers
112 views

### Is it correct to move x down in $2^x - 2^3 < 0$?

I have $2^x - 2^3 < 0$ and I think it's correct to conclude that $x - 3 < 0$ but a friend of mind disagree with me. I was wondering if there is such a property or axiom?
1answer
53 views

### Prove that $exp_{a}(\frac{p}{q}) = \sqrt[q]{a^{p}} \space \forall \space p,q \in \mathbb{Z}$

Prove that $exp_{a}(\frac{p}{q}) = \sqrt[q]{a^{p}} \space \forall \space p,q \in \mathbb{Z}$ with $q \geq 2$ I'm not sure how to approach this question. I was thinking through in induction with ...
3answers
93 views

### Modular Exponentiation 8^5^4 [duplicate]

I am trying to find the last digit of $8^{5^4}$ (or $8^{(5^4)}$, if you will) using modular exponentiation. What I know is that the value I want is: $8^{5^4} \mod 10$. Normally I would find a ...
3answers
80 views

### Modular Exponentiation 3^5^7

I am trying to find the last digit of $3^{5^7}$ (or $3^{(5^7)}$, if you will) using modular exponentiation. Here's what I've figured out: The value I want is $3^{5^7} \mod 10$. $5^n \mod 10 = 5$ (if ...
2answers
32 views

### Integrate a function using variable changement

I did the integraton using Wolfram alpha but I wonder if is it feasible using either integration by part or by substitution
4answers
64 views

### If $a^x=b$, then $x=$?

Stupid question, I know, but I couldn't remember nor find information by googling on how to find the exponent of $a$ that gives $b$ as the result. If $a^x=b$, then $x=log_a b$ but how do you find $x$?
1answer
65 views

### Deep Roots…:)

Why is it that powers with very small fractional or decimal exponents all tend to one? That is, for $x \ll 1$, $a^x \approx 1$, seemingly. True, or untrue? Can anyone offer more explanation? Thanks ...
3answers
52 views

### Power and exponential equation [duplicate]

So lately I came across this seemingly simple problem that I just can't get around. Solve this equation: $x^2 = 2^x$ I cannot do this algebraically, while I refuse to believe it is impossible to ...
0answers
57 views

1answer
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### Evaluating Exponents

Answer choices: 22 15 10 358 None of the above This is my solution:
3answers
73 views

### What $n^{\frac{1}{\log_2n}}$ means?

I was confused with about the $n^{\frac{1}{\log_2n}}$ expression. I am not sure how to make mathematical sense of it - i.e. express it in another way for easier understanding. I tried to plug in some ...
2answers
34 views

### Prove that I can always write a number a, a>0 as any number c, c>0 to the power of some number (a=c^x)

I'm very new to math, I'm sorry if my question is stupid. I started to study math by my own so I can study Computer Engeneering. I'm studying logarithms and I try to come up with simple proofs of the ...
1answer
57 views

### How to simplify logs and powers?

Is there any way to simplify $(\log a)^{\log b} = c$? And even this $(\log x)^y = z$? And also this $(\log m)(\log n) = p$ (which is essentially $\log m^{\log n} = p$) I was trying to simplify some ...
1answer
117 views

### zero raised to power zero in Church encoding

In Church encoding of the natural numbers in lambda calculus raising zero to the power zero gives the answer zero. Does anybody know of an encoding where the answer is 1?
2answers
4k views

### How do I reverse engineer this “power of”/exponent?

Take the following: (2)^3 = 8 I understand that this is 2 * 2 * 2 = 8 My question is how do I reverse engineer this if I ...
2answers
52 views

### Show if $0 \le a <b$ implies $0 \le a^{\frac{1}{n}}<b^{\frac{1}{n}}$

Given that $0\le a<b$ show that $0\leq a^{1/n}<b^{1/n}$ Is this proof by induction? Show it's correct for $n=1$ Assume true for $n=k$, then $0\leq a^{1/k}<b^{1/k}$ holds for some $k$, ...
2answers
35 views

### Why $(10^ab+c)^{4d+1}-c \mid 10$?

I came across the following equation: $$x=(10^ab+c)^{4d+1}-c$$ Why is $x$ a multiple of $10$ for any natural number values for $a$, $b$, $c$ and $d$? The only progress I made was that $a$ could be ...
0answers
25 views

### Exponentiation on order types

How is exponentiation defined on order types? We know that $2^\omega=\omega$. What is $2^{\omega^*}$? Is it $\omega^*$? $\eta$? $\lambda$? I'm guessing $\eta$, but I'm not sure. $\omega$ is the ...
2answers
25 views

### Using the distributive property to factor $(5^{-1}\cdot 5^x - 5^x - 5\cdot 5^x + 5^2\cdot 5^x)$

I can't seem to understand the distributive property. Take this: $$5^{-1}\cdot 5^x - 5^x - 5\cdot 5^x + 5^2\cdot 5^x$$ becoming this: $$5^x\left(\frac 15 - 1 - 5 +25\right)$$ Help? :D
3answers
40 views

### Beginner exponent/simplification question

Hey there I am having some trouble remembering all the old exponent rules and such, for example, $$\frac{1}{(6+7^n) ^3}$$ How can I simplify this? I know that (7^n)^3 is the same as (7^3n), but ...
2answers
46 views

### Modular exponentiation

How do you solve: $$5^{{9}{^{13}}^{17}} \equiv x\pmod {11}$$ I've been trying with this but no luck. I get to ${{9}{^{13}}^{17}} \equiv x\pmod {11}$ from $5^3 * 5^3 * 5^3 = 64 \equiv 9\pmod {11}$. ...
2answers
83 views

### Simplifying the exponential expression $e^{-4\ln x +8\ln y +2}$ [closed]

I'm totally stuck on this. Tried numerous sites for a decent explanation but can't find anything. Simplify the expression $$e^{-4\ln x +8\ln y +2}.$$ Thanks in advance.
1answer
65 views

### Quick methods to check perfect$4^{th}$,$5^{th}$, $6^{th}$ powers

Are there any quick modulus methods to check if a number could be a perfect power (4, 5, 6)? Preferably binary methods. For example, a perfect fourth power has to be $0, 1 \pmod{16}$ from a square ...
1answer
46 views

### What's the solution to this exponential system of equation?

What are the steps to solving a system of equations when $x$ and $y$ are exponents? But they have different base. Here is the problem. $5^x\times3^y=45$ $3^x\times5^y=75$
1answer
40 views

### Help solving simultaneous equation with powers

I am trying to solve the following equations: $0.5 = exp(-(3*c)^k)$ and $0.99 = exp(-(29*c)^k)$ I have used MATLAB to get the answers of $c = 0.21487$ and $k = 0.83471$ but I'd really like to know ...
3answers
49 views

### Are these exponential forms equal?

Is $(\frac1{\sqrt x})^{11}$? the same thing as $x^{\sqrt{11}}$ ? Basically what I'm asking is are those equivalent/the same?
1answer
52 views