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
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Rules of i ($\sqrt -1$) to a power

$i^{2014}$ power =? A. $i^{13}$ B. $ i ^{203}$ C. $i^{726}$ D. $i^{1993}$ E. $i^{2100}$ I don't understand the concept that powers of i repeat in fours and that "two powers of i are equal if their remainders are equal upon division by four". I…
user159778
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Evaluating the value of exponential expression

What is the value of: $\frac{2a}{a^{x-y}-1}+\frac{2a}{a^{y-x}-1}$ I tried this: $(\frac{a^{x-y}-1}{2a})^{-1}+(\frac{a^{y-x}-1}{2a})^{-1}$ $\frac{({a^{x-y}-1})^{-1}+({a^{y-x}-1})^{-1}}{2a^{-1}}$ But then I was stuck... any ideas?
Arda Arman
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exponentials with different base

What rule can I use when solving exponentials like this one $\frac {2^6 \cdot 5^8 \cdot 3}{100^3}$ I know how to solve exponentials when the bas number is the same with these formulas $x^m \cdot x^n = x^{(m+n)}$, $(x^m)^n = x^{(m \cdot n)}$, $\frac…
S4M1R
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exponent y=x^a sequences.

While analyzing square and cube functions, i found the following: for y=x^2 x=1, y=1 +3 x=2, y=4 +2 +5 x=3, y=9 +2 +7 x=4, y=16 +2 +9 x=5, y=25 increase of increase (well, how else should i say…
Jnsx
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Proving increasing function, base < 1, exponent increasing

For a fair lottery game where the odds of $1$ ticket winning are $1$ in $p$, where you can spend a total of $K$ dollars, and where you will spread your ticket purchases equally among $n$ draws, prove that the odds of winning at least once decrease…
user159407
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Fractional index fallacy

According to index laws, $(a^b)^c=a^{b\cdot c}=(a^c)^b$ However, if for example we have $a=-1, b=4, c=1/2$, then we get the equation: $$((-1)^4)^{1/2}=(-1)^2=((-1)^{1/2})^4$$ The first equation is equal to $1$, however, the last one is undefined.…
JAS
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Is there a way to write $2^3+2^2+2^1+2^0$ in short form or a better way?

I am doing a question and instead of going through phases solving the question I was wondering if I could do it all in one with a short equation. The question is about compound interest finding the future value. The number $2$ is used for ease of…
HelpAppreciated
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Is there a way to find x when x is an exponent?

I'm kind of stuck on how to solve the following. $10^x = 5^9$ Is there a method or a simple trick to find what is x?
kimnod
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Reciprocal of $7.5^{1-x}$

Ok my calculator tells me the reciprocal of $7.5^{1-x}$ is $0.1333\cdot7.5^x$. Can anyone explain the steps involved to get this manually? Is it along the line of the reciprocal of $7.5^1 + 7.5^{-x} = 0.13333 \cdot 7.5^{-x}$?
user3550682
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question on surds i already asked this question but the answer I got did not match the one in the book

$$\sqrt{ 3x }= x + \sqrt {3}$$ Give x in the form $$A \sqrt {B} + C $$ Can you show me how this is done step by step. The answer I have in the book is: $$\frac {1}{2} \sqrt{3} + \frac {3}{2} $$ this is where I got stuck: $$ \frac {x^2 +2x \sqrt{3}…
user133745
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If $(x^y)^z = x^{y\cdot z}$, why does $(-5)^{2^{0.5}}$not equal $(-5)^1$?

As shown by Wolfram Alpha, $(x^2)^{0.5}$ is equal to |x|, but if you tried to simplify it to $x^{2\times {0.5}}$, it would just be $x^1$, or $x$. Is there some unwritten rule about that distribution law that means you can't do it with fractional…
iirelu
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why is $(-64)^{2/3} =-16$ and not $16$?

It appears that taking the cube root of a negative number will yield a negative number, which when squared, will yield a positive number. But all the calculators and books I have seen show this particular problem yields a negative number. Any help…
Bill
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Relating to $( a ^ b ) + c = d$ and $( a ^ b ) - c = d$

Is there a way of deducing the smallest integer values for $a, b$ and $c$ that satisfy either $( a ^ b ) + c = d$ or $( a ^ b ) - c = d$ such that the addition $( a + b + c )$ is the smallest possible integer? I am wanting to do this for some very…
John Anthony Oliver
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Why the identity $ a^x = a^y \Longrightarrow x=y $ do not work for $a<0$?

All books that i am reading are telling that the identiy $ a^x = a^y \Longrightarrow x=y ; \,\,\,\,\, a \in \mathbb{R} - \{0,1\} $ $ \,\,\,\,$ also do not work for $a<0$, but, for example, if $ (-2)^2 = (-2)^x $, than for sure $x=2$. Than, the…
Voyager
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Representing Complex Exponentials with Real and Imaginary Parts

My confusion lies with this : http://www.wolframalpha.com/input/?i=modulus+%28cos%282+pi+r_1%29%2Bcos%282+pi+r_2%29%2Bi+%28sin%282+pi+r_1%29%2Bsin%282+pi+r_2%29%29%29+squared I was looking at alternate representations, and I was confused how to go…
user82004
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