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.

3996 questions
2
votes
2 answers

regarding exponents, how to interpret and use.

Some times in the books both of the below mentioned concepts are used interchangeably. Is there any reason for that? When to use -(2)^2 = -4 and (-2)^2. Explain with any useful examples.
VAR121
  • 307
  • 2
  • 3
  • 7
2
votes
2 answers

What is the square root of infinity and what is infinity^2?

In rigorous mathematics, how can one formalize the notion of a "square root of infinity", as well as that of "infinity squared"?
Mindozas
  • 107
  • 1
  • 1
  • 3
2
votes
0 answers

how to calulate the number of powers in finite field

For any finite field $F_q$ where $q$ is a prime power, I want to calulate the number of powers. Specifically, for 3$\le k \le q-1$, how to determine the size of the set: $\{x^k \mid x \in F_q\}$. Since the order of $F_q^{\times}$ is $q-1$, I only…
Zongxiang Yi
  • 1,079
  • 5
  • 15
2
votes
2 answers

When is a power real?

We know that the power $a^b$ is, if $b$ is not an integer, the product of a power and a root (example : $2^{2.5}=2^2\sqrt{2}$). But how do we know, if $a$ is negative, if $a^b$ is complex ? For example, $(-1)^{2.5}$ is complex ($i$), but…
PearlSek
  • 171
  • 2
  • 11
2
votes
2 answers

Rational complex exponent and proof that a matrix is unitary

I need to prove that a square matrix $A$ with elements $a_{jk}$ has orthogonal rows and columns: $$a_{jk} = exp\Big(\frac{2\pi i}{n}jk\Big)_{j,k=0}^{n-1}$$ It's possible to prove it by showing that $A$ multiplied by its conjugate transpose $B=A^*$…
Konstantin
  • 1,623
  • 2
  • 17
  • 36
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
  • 8
2
votes
3 answers

Provide an efficient algorithm for computing $(a + b \sqrt{3})^n$?

Provide an efficient algorithm that takes three integers as input and returns two integers that satisfy the condition. Only simple arithmetic operations are allowed (square root operation is not permitted). Input: $(a, b, n)$ where $a, b, n$ are…
Benjamin Z
  • 65
  • 4
2
votes
5 answers

Prove that $ 2/1!+4/3!+6/5!+8/7!+\dots = e$

Prove that $ 2/1!+4/3!+6/5!+8/7!+\dots = e$ My attempt: I googled the problem, and I found that $\sum_{(n=1)}^∞ \cfrac{2 n}{(2 n-1)!} = e$ I also found that $\sum _{n=0}^{\infty \:}\cfrac{2n+2}{\left(2n+1\right)!}$ is equal to $e$. How can I…
2
votes
3 answers

Calculate limit involving exponents

Calculate: $$\lim_{x \rightarrow 0} \frac{(1+2x)^{\frac{1}{x}} - (1+x)^{\frac{2}{x}}}{x}$$ I've tried to calculate the limit of each term of the subtraction: $$\lim_{x \rightarrow 0} \frac{(1+2x)^{\frac{1}{x}}}{x}$$ $$\lim_{x \rightarrow 0}…
George R.
  • 2,763
  • 8
  • 15
2
votes
3 answers

Expressing $b^x$ as $e^{x \ln b}$

Can anyone explain the following equality please? $$b^x = e^{x \ln (b)}$$ I've verified it with values but I can't think of what the proof for this would be.
2
votes
1 answer

Help in solving exponential equation

Solve the following equation: $$\frac{8^x + 27^x}{12^x + 18^x} = \frac{7}{6}$$ All I managed to do is rewrite the given equation in a simpler form: $$\frac{4^x}{6^x + 9^x} + \frac{9^x}{6^x + 4^x} = \frac{7}{6}$$ I don't know what should be done…
George R.
  • 2,763
  • 8
  • 15
2
votes
1 answer

Why is the square root of the square of a negative number /variable always positive?

For negative values of x, $(x^2)^{0.5}$ is $\pm x$. For negative values of x, $(x^{0.5})^2$ is undefined. Why do textbooks insist the following: for negative values of $x$, $\sqrt{x^2} = |x|$? Surely the correct way to think about this is that for…
2
votes
1 answer

Proof that any rational can be bound by two powers of another rational

Let $\epsilon \in \mathbb{Q}, \epsilon>0$ and $a\in\mathbb {Q}, a>1$, prove that exists $n_1,n_2 \in\mathbb {N} $ such that $$a^{-n_2}<\epsilon
2
votes
2 answers

Is my $0^0 = 1$ proof correct?

Is the following proof that $0^0 = 1$ correct? Let $k = 0^0$ (assuming it has a meaningful value) Observe that $k = 0^0 = 0^{n0} = (0^0)^n = k^n$, for all integers $n$. $\implies k^n - k = 0$, for all $n \in \mathbb Z$. $\implies k^n - k = 0$ and $k…
A.Abbas
  • 71
  • 6
2
votes
0 answers

Rational numbers with odd denominator

Notation. By $\mathbb{Z}_{(2)}$, I mean the localization of $\mathbb{Z}$ at the prime ideal $(2).$ So basically, this is obtained by adjoining a multiplicative inverse for every positive prime number distinct from $2$. Equivalently, we can think of…
1 2 3
99
100