Mathematics Stack Exchange
Mathematics Stack Exchange

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
2
votes
2 answers

Prove that $\sqrt[kn]{a^{km}}=\sqrt[n]{a^m}$

$a>0$. I'm trying to prove that $\sqrt[kn]{a^{km}}=\sqrt[n]{a^ m}$. Here's what I tried to do, but I think it's useless. $\sqrt[kn]{a^{km}}=\sqrt[kn]{(a^m)^k} = ((a^m)^k)^{1/kn} = \Bigr(\bigr((a^m)^k\bigr)^{1/k}\Bigr)^{1/n} = (a^{m/n})^k$.
David
  • 317
2
votes
2 answers

Finding all expressions of a number as an exponentiation

Is there a way to systematically determine all the ways a number can be expressed as an exponentiation of two natural numbers? For example, 64 = 64^1 = 8^2 = 4^3 = 2^6 , but how would I determine this without brute-forcing through roots or…
Tobl
  • 123
2
votes
3 answers

Evaluate $5+4\cdot 5+4\cdot5^2+4\cdot5^3+4\cdot5^4+4\cdot5^5$

Evaluate $$5+4\cdot 5+4\cdot5^2+4\cdot5^3+4\cdot5^4+4\cdot5^5.$$ The options are $5^6$, $5^7$, $5^8$, $5^9$, $5^{10}$. I'm new to this site. I came across this question in an Olympiad foundation site. I have no idea how to solve it. Can I get…
Swayam Pattanaik
  • 21
2
votes
3 answers

If $10^a = 8$ and $10^b=12$ and $10^x = 6$ what is in a and b

The title pretty much sums it up. If we had an equation $10^a = 8$ and $10^b=12$ what is $x$ in terms of $a$ and $b$ if $10^x =6$. I feel like the answer is simple, but yet I only got so far. Thank you
John Hon
  • 473
2
votes
2 answers

Figuring out $x^n$

Exponents are used to represent multiplying by a number over and over. but big numbers, like $6^8$ are hard to calculate. is there any simple way to calculate big numbers of the form $x^y$? ($y>0$ and is whole)
Alexander Day
  • 885
2
votes
1 answer

exponent problem

Can anyone answer this question? $$3^{n+2} + (3^{n+3} - 3^{n+1})= ?$$ I really want to know how to solve this one, our solution can't seem to agree with the problem's answer thank you!
yllika
  • 33
2
votes
2 answers

When does $(-1)^x$ give reals solutions?

For any real x, how can I know if $(-1)^x\in\mathbb R$? We easely guess that if x is an integer, $(-1)^x$ has a real solution. If $x = \frac{1}{2n+1}$ with $n\in\mathbb Z$, it also gives a real solution. So then I wonder if... is that all? Did I…
Quentin Januel
  • 21
2
votes
6 answers

Identity proof $\dfrac{x^n-y^n}{x-y} = \sum_{k=1}^{n} x^{n-k}y^{k-1}$

In a proof from a textbook they use the following identity (without proof): $$\dfrac{x^n-y^n}{x-y} = \sum_{k=1}^{n} x^{n-k}y^{k-1}$$ Is there an easy way to prove the above? I suppose maybe an induction proof will be appropriate, but I would…
whoisitnow
  • 131
2
votes
3 answers

How would we solve $x^2 = 2^x$ mathematically rather than logically?

We have the equation in $x$: $$x^2 = 2^x$$ We know that, by logic, if we equate the bases and the powers separately, we get $x=2$ in both the cases and thus we conclude that $2$ is the root of the equation. But what if we don't apply that logic. Is…
Mihir Chaturvedi
  • 603
  • 1
  • 5
  • 11
2
votes
3 answers

I want to Find $3^{2x} + 7^{\frac{1}{x}}$ when $7^{x+1} = 21^{x}$

I figured that \begin{align*}7^{x+1} &= 21^{x}\\ 7^{x+1} &= 3^{x} \times 7^{x} \\ 7 &= 3^{x} \end{align*} but I can't go further at the moment.
SarpSTA
  • 467
2
votes
1 answer

Compare between exponents

which one will be larger, $99^{99}-99^{98}$ or $99^{98}$ I could not find any exponent properties that will help solving this.
user3188039
  • 155
2
votes
2 answers

If $x^a$ is $x$ multiplied $a$ times, then how does $x^{-1}$ make sense?

What is the meaning of $x$ raised to any non-positive value? We know that $x^{-a} = \dfrac{1}{x^a}$ and $x^0 = 1$, but where does that come from? What is the proof? Why is this true? What about $x$ raised to a fraction, like say $\frac{1}{3}$? How…
jimpix
  • 211
2
votes
2 answers

Exponentiation with negative base and properties

I was working on some exponentiation, mostly with rational bases and exponents. And I stuck with something looks so simple: $(-2)^{\frac{1}{2}}$ I know this must be $\sqrt{-2}$, therfore must be imaginary number. However, when I applied some…
Harry Hong
  • 21
2
votes
1 answer

$p^q > q^p$ 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 greater was to put them in to my calculator (casio…
george
  • 144
2
votes
1 answer

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 point?
exp8j
  • 587
1 2 3
…
35 36