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

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0
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2answers
21 views

$x^a$ not defined in $\Re$ when x negative?

I know that by definition $x^a$ = exp(alogx) which is not defined when x not in $\Re^∗_+$ Does it mean that for example $-1^3$ not defined for real numbers ?
1
vote
2answers
86 views

Find root of $x\cdot 2^x - 1$ function

Is it possible to solve the equation: $x2^x - 1 = 0$ without using the function graph? According to the function plot, its root is around 0.6. I need to get the numeric value of the function's ...
0
votes
4answers
78 views

How to solve $(\frac{1}{x})^2x=e^6$? [closed]

I need to solve this: $$\left(\frac{1}{x}\right)^2x=e^6$$ I know it's equal $$\ x=\frac{1}{e^6}$$ But how?
0
votes
2answers
31 views

What is the minimum $k$ such that $\sum_{i = 1}^{k}2^{a_i} = \sum_{j = 1}^{n}2^{w_j}$

Suppose you have a sequence of numbers $S = (w_1,...,w_n)$ and you want to know the minimum $k$ such that $2^{a_1}+...+2^{a_k} = 2^{w_1}+...+2^{w_n} = T$. I'm told that the minimum $k$ is equal to the ...
2
votes
2answers
33 views

relation between exponent values and solution

If $a,b,c$ and $d$ are positive integers such that $a^5=b^6$ and $c^3=d^4$ and $d-a=61$, then find the smallest value for $c-b$. $d-a=61$, so $d=a+61$ or $a=61-d$. But then how to substitute ...
1
vote
2answers
26 views

By which factor does b increase if a is multiplied with 2?

Consider $$2a^2 = 3b^3.$$ By which factor does b increase if a is multiplied by 2? Answer is 4. Can anyone explain it to me?
2
votes
1answer
59 views

Express an exponential integral in hypergeometric form

I am new to hypergeometric function. I am trying to express this: $$\int_{0}^{\infty}e^{-ax^k+bx}dx$$ in a hypergeometric form. I have read some reference, but I don't get it how to cope with this ...
2
votes
1answer
44 views

If $2^n=3^m$ and both $n,m$ are integers, which one is greater?

If $n$ and $m$ are whole numbers. $$2^n = 3^m$$ Which one is bigger, $n$ or $m$? I don't know how to approach this question.
0
votes
3answers
74 views

Let $a,b,n$ be any integers, show that: $x^{n}\cdot{y^{n}}=(xy)^{n}$

I know the rule of adding exponents while multiplying same integers: $$x^{a}\cdot{x^{b}}=x^{a+b}$$ but that doesn't seem to help, in this case. An elementary proof would be appreciated. Regards
1
vote
0answers
18 views

Exponential of the product of a scalar with a complex symmetric $2 \times 2$ matrix

Say I want to compute $e^{t M}$, for $M=\left[\begin{array}{cc} a & b\\ b & a \end{array}\right]$ with $a, b \in \mathbb{C}$ and $t \in \mathbb{R}$. Can I absorb $t$ into $M$ and do as ...
1
vote
3answers
44 views

simplify a log expression

I met a problem, I don't know if this term can be simplified properly? $$e^{ (\ln ax^{b})^{c}}$$ since the ln term with power of c is hard to cope with, thanks for any help!
0
votes
1answer
37 views

Exponents with bivectors

According to the Wikipedia page on bivectors: ...if $B$ is a bivector, then the rotor $R$ is $e^{B/2}$ and rotations are generated [by] $v'=RvR^{-1}$. But how do you take an exponent between a ...
1
vote
1answer
28 views

Converting $2^{16x} \equiv 2^6\text{ (mod 19)}$ to $16x\equiv 6\text{ (mod 18)}$

In my number theory book, I see the congruence $2^{16x} \equiv 2^6\text{ (mod 19)}$ being converted to $16x\equiv 6\text{ (mod 18)}$. My question is: Why is this allowed and what are the general ...
3
votes
8answers
113 views

What do you get when you take the square root of a negative imaginary number?

Simply, what do I get when I take the square root of negative imaginary number? But, it cant be an imaginary number since the answer to the $2^{\mathrm{nd}}$ power must equal the original negative ...
1
vote
1answer
55 views

Exponential integral problem

I am trying to solve this $\int_{0}^{\infty} e^{-ax^k-bx}dx$ I tried to solve this by considering this as an laplace transform of $e^{-ax^k}$ ,but I cannot find it in a laplace table or in online ...
0
votes
1answer
17 views

Repeating units digits - is this pattern meaningful or random?

I'm studying for the GRE, which often has problems involving the units digit of $n^m$, for some integers $n$ and $m$, e.g. "What is the units digit of $2^{300}$. I decided to create a table to see if ...
4
votes
1answer
96 views

If $x^{x^4} = 4$, what is the value of $x^{x^2} + x^{x^8}$? [duplicate]

If $x^{x^4} = 4$, what is the value of $x^{x^2} + x^{x^8}$ ? I can find by trial and error, that $x=\sqrt 2$. But, what is the general process to answer questions like this?
0
votes
3answers
50 views

Why does a root only have a positive output? [duplicate]

Let's say I am solving an equation, and end up with this: x^2 = 16 The solutions will be x=4 or x=-4 That makes sense. But when I have this: x = √16 The only solution is x=4 ...
8
votes
4answers
105 views

What is $(-1)^{3/2}$?

Which one is the right way? $$(-1)^{\frac32}=(-1)^{1+\frac12}=-1\times \sqrt{-1}=-i$$ Or, $$(-1)^{\frac32}= \left((-1)^{\frac12}\right)^{3}= i^3=-i$$ Or, ...
4
votes
2answers
66 views

If $2^x/\sqrt{x}=n$, how does $x$ grow asymptotically?

If $\dfrac{2^x}{\sqrt{x}}=n$ where $x\rightarrow\infty$ as $n\rightarrow\infty$, how does $x$ grow asymptotically in terms of $n$? We know that it grows faster than $\log n$, because $\dfrac{2^{\log ...
1
vote
1answer
68 views

If $x^x=n$, how does $x$ grow asymptotically?

If $x^x=n$, how does $x$ grow asymptotically in terms of $n$? We know that it grows slower than $\log n$, because $\log n^{\log n}>e^{\log n}=n$. But it grows faster than $\log \log n$, because ...
5
votes
2answers
144 views

For which orthogonal matrices does the matrix exponential converge?

Part (a) For which 2×2 orthogonal matrices A does $\large e^A=I+\frac{A^1}{1!}+\frac{A^2}{2!}+…$ converge? Part(b) For what A does the series converge to an orthogonal matrix? My work: Let A ...
0
votes
0answers
31 views

Simplify an exponential integral

I want to solve this : $\int_{0}^{\infty}\frac{1-e^{-x}}{x}e^{-\lambda x^{\alpha}}dx$ I had asked a similar problems yesterday, like this ,but we get power of alpha of x and I cannot use Frullani's ...
0
votes
1answer
47 views

If $x= 2$ find the value of $3^{x+2} + 5^x - 7^{x-1}$

If $x= 2$ find the value of $3^{x+2} + 5^x - 7^{x-1}$ For this question do I simply replace $x$ as $2$ so the final answer would be $99$ or is there a more complicated procedure?
1
vote
4answers
43 views

Question about the identity of $2^{2^n}$

Is this identity true: $2^{2^n}= 4^n$? I believe this is true as far as I know. Sorry this is the only place to ask. Is there another identity for $2^{2^n}$ which I can simplify to?
0
votes
2answers
34 views

Definition of Zeroth Power [duplicate]

What is the definition of raising a number to the zeroth power ($x^0$)? I know that many people say that "anything raised to the zeroth power is one" but this is clearly not true since $0^0$ is ...
2
votes
1answer
73 views

Simply this exponential integral

I am trying to simplify of this: $$\int_{0}^{\infty} \frac{1-e^{-x}}{x}e^{-\lambda x}\,dx.$$ Maybe I should separate these equation into two exponential integral function? But it will ended up with ...
2
votes
2answers
41 views

Solve the inequation for $x$

Solve for $x$ : $ (x-1)^{2005} x^{2006} (x+1)^{2007} \le 0 $ I tried cases like : $ x-1 \le 0 $ , $ x \le 0 $ , $ x+1 \le 0 $
2
votes
1answer
69 views

Fermats little theorem, $p$ is not a prime number [closed]

Calculate the remainder $$ r \equiv 37^{877} \bmod{323} $$ I don't know how to follow this up since $323$ is not a prime number.
1
vote
2answers
44 views

Best way to solve $24^{100} \times 1.5^{50} \times 12^{-149}$

I think I could solve this but I would like to know the best way to do it with the least amount of calculations
1
vote
1answer
66 views

Finding the value of $x^{x^2}+x^{x^6}$ given that $x^{x^4}=4$

If $x^{x^4}=4$; Then what is the value of $x^{x^2}+x^{x^6}$ ; I just want a hint like from where should I start. (other than using a trial and error to find that $x=\sqrt{2}$ satisfies this and now ...
1
vote
1answer
49 views

How would I prove $\lceil \frac{\binom{2^{n+1}}{2}}{2^{n+1}}\rceil = 2^n$ for all natural numbers n? [closed]

Any leads on how to prove $\lceil \frac{\binom{2^{n+1}}{2}}{2^{n+1}}\rceil = 2^n$?
1
vote
2answers
40 views

Computing $4^{m+1} \cdot 9^{n-1}$ in terms of $2^m \cdot 3^n$ [closed]

So I got this math question that I have to do. Unfortunately I don't understand a thing. The question is: If $2^m \cdot 3^n = a$, what is $4^{m+1} \cdot 9^{n-1}$? I will be grateful for any and ...
-1
votes
1answer
44 views

Solving pre logarithmic problems [closed]

How can I solve $2^{6x}$ when $8^x=3$?
1
vote
2answers
59 views

How to derive these Lie Series formulas

Relates issues: How to properly apply the Lie Series Exponential of a function times derivative In my old notes about Lie groups and/or operator calculus, I've encountered the following formulas: $$ ...
0
votes
4answers
70 views

Finding the real value of x

I replaced the numbers with variables so i let $y=2$ and $z=12$ and i got the following after simplifying the left side: $$2^{144}-\frac{1}{2^{144}}=8^x-8^{-x}$$ but i'm not seeing how this can ...
0
votes
1answer
188 views

Find $10^{5^{101}}$ modulo $21$

Most of the solutions to this question goes on by stating $10^{5^{101}}$ as $10^{505}$. But shouldn't this be thought as $5^{101}$, not $(10)^{5\cdot 101}$? Well if I do solve it as $10^{505}$, I get ...
2
votes
5answers
123 views

Finding the final digit

What is the final digit in $\left(\cdots \left(\left(\left(7^7\right)^7\right)^7\right)^7\cdots\right)^7$, where the $7^{\text{th}}$ power is taken $100$ times? So I'm trying to see how to do ...
0
votes
2answers
79 views

Are the logarithm rules and exponentiation rules (e.g. $a^{x+y}=a^xa^y$) axioms when talking about real numbers?

Are the logarithm rules and exponentiation rules (e.g. $a^{x+y}=a^xa^y$) axioms when talking about real numbers? I know many of them can be proved via induction for integers, but no professor has ...
0
votes
2answers
36 views

Exponent of Sum property

I was reading the exponent Combination law in proofwiki.org and got confused in one part of the proof. The proof is as follows: Let $a \in R{> 0}$ Let $x, y \in R$ Let $a^x$ be defined as $a$ to ...
0
votes
1answer
39 views

Solution $(x,y)$ to $x^y=y^x=a$, $a>1$ being real.

I seem to have found a method to compute the solution $(x,y)$ to the equation, $x^y=y^x=a$ $a\geq 1$ where $a$ is real, by using limits. But I don't know if this is something new. Does there already ...
1
vote
1answer
61 views

x to the power of an irrational number

If one were to graph the function of $$ f(x) = x^e $$ How would this look? (With explanation as to why) Particularly in the case of negative x values.
0
votes
0answers
66 views

How do we know $\pi$ cannot be expressed a root [duplicate]

In other words, is there a proof that $\pi^a\neq b$ where $a,b\in \mathbb{Z}$?
0
votes
1answer
50 views

Exponents math problem (Pre-Algebra) [closed]

(1) $x^m\cdot x^n = x^{m+n}$ List all the possible cases of whole numbers $m$ and $n$ for identity (1) more precisely, when $m > 0$ and $n > 0$, we already know that (1) is correct. What ...
0
votes
2answers
60 views

sum of the digits of the number $(5^{2015})(2^{2018})$

What is the sum of the digits of the number $(5^{2015})(2^{2018})$ So i am guessing, i have to find out the product of $(5^{2015})(2^{2018})$ and add each digit of the product. The question is how ...
3
votes
3answers
337 views

In a set A ={1,2,3,4,5,7,8,10,11,14,17,18} how many subsets of A contain only Odd numbers?

I know the answer is 64 from 2^6 but I may just not be great at logic or fully know the definition of a power because I am confused on the reasoning why. The reasoning I heard is you have 2 choices: ...
0
votes
0answers
26 views

How to find last 9 digits of a large exponent?

I was doing some programming problem where I had to find the sum of last 9 digits of very large exponents. I can't even think of calculating that big powers, but I have to find last 9 digits of the ...
1
vote
3answers
61 views

Formula for $A^n$ where $n \in \{1, \ 2, \ \cdots \ \}$ for the matrix $A = \begin{bmatrix} 1 && b \\ 0 && 1 \end{bmatrix}$

Formula for $A^n$ where $n \in \{1, \ 2, \ \cdots \ \}$ for the matrix $A = \begin{bmatrix} 1 && b \\ 0 && 1 \end{bmatrix}.$ Please help with the question if you can, it is for my ...
0
votes
3answers
85 views

limit of $\frac{x^x -1}{x}$ as x tends to zero

I was wondering if i could get any help with the following: $$ \lim_{x \rightarrow 0^{+}} \frac{x^x -1}{x} $$. thank you. My attempt: $$ \lim = \frac{e^{x \ln(x)} - 1}{x} = \lim ( x \ln (x)( x ...
4
votes
4answers
92 views

Solve x when it is an exponent…

Please help me solve: $$5.5^{2x}-6.5^x +1 = 0$$ I really don't think it is possible at all. They wanted me to do it without an calculator. By the way I am new to the forum so I don't know if I have ...