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

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0
votes
2answers
36 views

Is it possible to convert $y = a^x + b^x$ to the form of $y = a \cdot b^x$?

I don't think this is possible, but I wanted to ask people who know more than me. Is it possible to convert $y = a^x + b^x$ to the form of $y = a \cdot b^x$?
7
votes
4answers
173 views

Find the value of $x$ if $x^{x^4} =4$

Find the value of $x$ if $x^{x^4}=4$. Given options are $2^{1/2}$ $-2^{1/2} $ Both 1. & 2 None of the above From option verification, we get option 3. as correct one. But is there any real ...
-1
votes
2answers
42 views

$((-a)^3)^{1/2}\ne(-a)^{3*1/2}?$

The original problem is $\sqrt{(-a)^{3}}\sqrt{(-a)}$ I attempted to solve this as the following way: $\sqrt{(-a)^{3}}\sqrt{(-a)} = (-a)^{\frac{3}{2}}(-a)^{\frac{1}{2}}=(-a)^{2}=a^{2}$ However, I ...
7
votes
2answers
265 views

Fermat's Little Theorem and Euler's Theorem

I'm having trouble understanding clever applications of Fermat's Little Theorem and its generalization, Euler's Theorem. I already understand the derivation of both, but I can't think of ways to use ...
6
votes
5answers
295 views

Solve $4 \times2^x+3^x=5^x$ without any sort of calculator

Is there a way i can solve the following equation only by using high school mathematics? $$4 \times2^x+3^x=5^x$$ I tried writing $5$ as $2+3$ but didn't get any result. After that i tried to divide ...
2
votes
5answers
84 views

When is $\sqrt{x/y^2}$ equal to $\sqrt{x}/y$?

The solution to the quadratics is given by $r = -\dfrac{b}{2a}\pm\sqrt{\dfrac{b^2-4ac}{4a^2}}$, which is shortened to $r = -\dfrac{b}{2a}\pm\dfrac{\sqrt{b^2-4ac}}{2a}$, but I'm wondering how if this ...
1
vote
4answers
69 views

Why is $-32^{\frac{1}{5}} = 2$

When you factorize $-32$, you get: $-32 = (-16) \cdot 2$ $-16 = (-8) \cdot 2$ $-8 = (-4) \cdot 2$ $-4 = (-2) \cdot 2$ $-32^{\frac{1}{5}} = -2$ The reason I am asking is because you get $-4 = -2 \...
-3
votes
2answers
67 views

How to solve $x^3+y^3=z^4$? [on hold]

Any three positive integers that are greater than 2 satisfying $x^3+y^3=z^4$ . You can not have two or more of the same number.
1
vote
0answers
62 views

How does $3n+1$ change the proximity of $n$ to a power of two?

This is part of an attempt to prove Collatz's conjecture. I proved a modification of Collatz's conjecture, where instead of $3n+1$ if $n$ is odd, you do $n+1$. In Collatz's conjecture, if you get to a ...
-3
votes
2answers
39 views

What mean $x^\frac{a}{b}$ where $a,b \in\mathbb{R}$? [on hold]

What mean $x^\frac{a}{b}$ where $a,b \in\mathbb{R}$? a) $\sqrt[b] {x^a}$ b) ($\sqrt[b] x)^a$ Thanks to all.
2
votes
4answers
172 views

“Exponential Madness” (Gauss's challenge)

From Euler's identity, we see that $e^{i\pi}=-1$ $\Rightarrow e^{2ik\pi}=1$ [squaring both sides]. This equation surely holds for all integers $k$. EDIT: From the second equation we get $e^{1+...
1
vote
2answers
25 views

Is there an imaginary Exponent with $a^x=-b$ while a,b>0

Is there an imaginary Exponent $x$ with $a^x=-b$ while a,b>0 ? An where could something like this be used?
0
votes
2answers
49 views

$e^{e^{10^{10^{2.8}}}}$ changing $e$ with $10$

From Numberphile $$e^{e^{10^{10^{2.8}}}}$$ changing $e$ with $10$, is there a way to change only the top most number while keeping all other numbers 10? i.e what is x in : $$e^{e^{10^{10^{2.8}}}} = ...
0
votes
1answer
34 views

Calculating the last digit of exponent

I need to calculate the last digit of $723^n$.(For every positive integer $n$). If it was to calculate the last digit of $a^b$ when I know the value of $a$ and $b$,then it was easy- for example,If I ...
9
votes
5answers
289 views

Which number is greater, $11^{11}$ or $9^{12}$?

Which number is greater than $11^{11}$ or $9^{12}$? My work so far: $11^{11}=285311670611>9^{12}=282429536481$. But to verify the validity of equality should be in the range of easily ...
2
votes
4answers
53 views

Derivative of exponent

Looking to solve : $$ \frac{d}{dx}[2^{0.5x}]$$ The multiplication and X value in the exponent is confusing me. Help? Thanks!
0
votes
0answers
36 views

question concerning tetration to infinity.

So I read on-line that \begin{equation} x^{x^{x^{x^{x^{x}}}}}=2 \end{equation} where number of x goes to infinity can be solved by solving \begin{equation} x^{2}=2 \end{equation} so by the same logic \...
3
votes
1answer
123 views

Meaning of imaginary part of $\int_0^6 e^{x^3} dx$

My question is the title itself. How can it be possible that integral of real numbers can have an imaginary part?
1
vote
1answer
33 views

Seeming Contradiction With Complex Exponents

I was fooling around while waiting for a page to load and came across the following "contradiction". Let $x=(-1)^{i}$. Then $x^{i}=(-1)^{i\cdot i}=(-1)^{-1}=-1$. Thus, $x=\left(x^{i}\right)^{-i}=(-1)^...
0
votes
2answers
20 views

Strong Induction issue

I am trying to prove a statement using strong induction but I seem to be getting stuck. I don't know if did something wrong or I am just not recognizing an opportunity for factoring/how to factor ...
-1
votes
1answer
37 views

How do I get negative exponent? [closed]

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 ...
2
votes
2answers
41 views

Logarithm of >2 numbers

I am learning logarithms and i found that $log(a*b) = log(a)+log(b)$ I tried to apply the same principle for three numbers like $log(a*b*c) = log(a)+log(b)+log(c)$ but it didn't work as i expected. ...
1
vote
1answer
75 views

Maybe this inequality holds? $x!-y!>x^n$?

Let $x,y,n$ be postive integers such that $x\ge 2y,y>n,n>3$ I conjectured that $$\color{red}{x!-y!\ge x^n}$$ Now, I claim that $$\color{red}{x!-y!=y![x(x-1)(x-2)\cdots(y+1)-1]\ge (n+1)![x(x-1)(...
0
votes
1answer
92 views

What is the value of $e^{-10000}$?

What is the value of $e^{-10000}$? We know that the function $e$ does not attain value $0$ anymore. But in R and Matlab the value of $e^{-10000}$ is given as $0$ which is not correct anymore. I ...
0
votes
2answers
119 views
0
votes
0answers
48 views

Interpreting $\log_2\left(\frac{1}{0}\right)$ and $\log_2\left(\frac{0}{0}\right)$

I'm hoping someone can confirm what I've done is correct. I am working with biological datasets ... about 100 RNA-Seq datasets and I'm trying to analyze the relative up or down regulation of genes. ...
0
votes
3answers
50 views

Why is $\sqrt{x/x^{-1}}$ OR $\sqrt{x/{1/x}}$ = $\lvert x\rvert$ and not just x

I have this task: Find equal expression to square root of fraction of x and its inverted value (this is translated from my mother tongue so I'm sorry if I've used incorrect terms). Anyway the starting ...
3
votes
4answers
72 views

Why is the inverse tangent function not equivalent to the reciprocal of the tangent function?

I know that $$ {\tan}^2\theta = {\tan}\theta \cdot {\tan}\theta $$ So I guess the superscript on a trigonometric function is just like a normal superscript: $$ {\tan}^x\theta = {({\tan}\theta)}^{x} ...
4
votes
5answers
172 views

Find $3^{333} + 7^{777}\pmod{ 50}$

As title say, I need to find remainder of these to numbers. I know that here is plenty of similar questions, but non of these gives me right explanation. I always get stuck at some point (mostly right ...
0
votes
1answer
36 views

How to expand and solve using exponent laws and algebra?

How do you solve this question, specifically between lines 2 and 3 in the image? The intuition, expanding and math behind it would be much appreciated! Question needing solved
1
vote
2answers
53 views

Express $(100^3)^5$ with a base of $10$

Express $(100^3)^5$ with a base of $10$. I don't get this.
1
vote
2answers
45 views

How would one solve this exponent problem without a calculator?

This is a problem on a non-calculator portion of an algebra 2 honors worksheet. $$2^{24}=(3x-1)^8$$
0
votes
3answers
86 views

Simplify the expression $(a^2)^5(2x^2)^4\over2^5(a^3)^3(x^3)^2$

Can anybody please provide a step by step solution to the following expression ? $$(a^2)^5(2x^2)^4\over2^5(a^3)^3(x^3)^2$$
3
votes
4answers
387 views

Exponential Equations with Fractions

I have had some issues with the following two equations: $$ \frac{3^{n-2}}{9^{1-n}}=9$$ $$\frac{5^{3n-3}}{25^{n-3}}=125$$ If anyone could work them out step by step that would be awesome. I ...
2
votes
0answers
31 views

Exponential of a symmetric, tridiagonal matrix

Is there an analytic result for the exponential of a symmetric, tridiagonal matrix (the diagonal can be zero, if this helps). Moreover, if simplifies the result, the sub-diagonal and supra-diagonal (...
-1
votes
2answers
26 views

Exponentiation - a to the i power

So if, $i=\sqrt{-1}$ and, $i=(-1)^\frac{1}{2}$ then, $a^i=a^{-1^\frac{1}{2}}=(\frac{1}{a})^{\frac{1}{2}}=\sqrt{\frac{1}{a}}$ ???
1
vote
0answers
13 views

The multivalued behaviour of complex exponential $z^\lambda$

On Gustav Doetch's Introduction to the Theory and Application of the Laplace Transform, it says: The power series $\sum_{n=0}^\infty a_nz^n$ converges on a circular disc. Replacing the integers $n$...
0
votes
1answer
24 views

Does equivariant property commute with matrix exponential?

Given a vector space $V$ and endomorphism $M : V \to V$ we can define the exponential of $M$: $\exp(M) = \sum_{k=0}^{\infty} \frac{1}{k!}M^k$. This has the property that it commutes with conjugation ...
2
votes
1answer
33 views

$p^q > q^p$ For what values of p and q does this hold true?

I saw a question on the internet the other day that asked which of two statements was larger. The numbers were 2^999 and 999^2. With the first being the greater number. One way to test which was ...
1
vote
1answer
48 views

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 ...
2
votes
1answer
46 views

Is $x^x$ in the same asymptotic growth class as an exponential function?

I see that for any natural number $a$, $\lim_{x\to\infty} \tfrac{x^x}{a^x}$ approaches $\infty$, so the limit does not exist. So is this function have a different big-O than $O(a^x)$, for example? So ...
2
votes
1answer
30 views

Raising to power geometrically

On a straight line with marked origin 0 and unit 1, two points x and y are given. Is it possible, by any finite method, to geometrically define x^y if the given y is not a rational but an arbitrary ...
2
votes
1answer
64 views

Fifth last digit of a huge number

How can I find the fifth last digit of $5^{5^{5^{5^5}}}$? I tried to evaluate $5^{5^{5^{5^5}}}\pmod {100000}$. But the exponent is so huge that I'm unable to evaluate it. Also, $(5,100000)=5$ , so $5$ ...
1
vote
1answer
92 views

Solve the system of equations $x^y=y^x$

Solve the system of equations $$ x^y=y^x \\ a^x=b^y $$ I could not solve this despite many tries
0
votes
0answers
44 views

Studying mathematics concretely, axiomatically & philosophically

I see mathematics as structures made up of numbers & shapes and actions done with them, like additions-multiplications-exponentiations, or divisions in pieces, translations, rotations, or some ...
1
vote
6answers
98 views

Simplifying $3^{-4}$ to $1/81$

How would you solve for $3^{-4}$? I know the answer is $1/81$ but I can't work out how you get there with this one.
0
votes
1answer
37 views

$4^{3a-1}-5^{2b-3}=0$ find a in terms of $b$

If $4^{3a-1}-5^{2b-3}=0$ then find a using $b$. My Attempt:we know that $2^{6a-2}=5^{2b-3}$ with this way we can find a value for $a$ and $b$ if both sides are zero if we can find another value for $...
0
votes
2answers
16 views

Image: proof re fractional exponents

Can someone help me prove that $(a^m)^{1/n} = (a^{1/n})^m$ per the textbook excerpt captured in the image?
0
votes
2answers
34 views

log to exponential form, but with number in front of log

So I understand how to put a log equation into exponential form. For example, $y = \log_2(x)$ is $2^y = x.$ However, I don't understand what to do when there is a number in front of $\log$, such as $...
1
vote
2answers
73 views

$8^a=3$ and $3^b=5$ and $10^c=5$ then find $c$ in terms of $a$ and $b$.

if $8^a=3$ and $3^b=5$ and $10^c=5$ then find $c$ using $a$ and $b$. My Attempt: if $8^a=3$ and $3^b=5$ then we can say that $8^{ab}=5$ and then we have $2^{3ab}=10^c$ but i cant solve this equation....