# Tagged Questions

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

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### What is the solution to the equation $9^x - 6^x - 2\cdot 4^x = 0$?

I want to solve: $$9^x - 6^x - 2\cdot 4^x = 0$$ I was able to get to the equation below by substituting $a$ for $3^x$ and $b$ for $2^x$: $$a^2 - ab - 2b^2 = 0$$ And then I tried \begin{align}x ...
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### Check whether a number could expressed as power of another two numbers [duplicate]

I found in many places how to find whther a number could be expressed as power of 2. What I need to know is, if a number is given whther that number could be expressed as a number raised to another. ...
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### Sum of super exponentiation

$f(x,n)=x^{2^{1}}+x^{2^{2}}+x^{2^{3}}+...+x^{2^{n}}$ Example: $f(2,10)$ mod $1000000007$ = $180974681$ Calculate $\sum_{x=2}^{10^{7}} f(x,10^{18})$ mod $1000000007$. We know that $a^{b^{c}}$ mod ...
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### Harmonic Mean Solution

The harmonic mean of two positive numbers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean ...
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### How can one solve $1^x=2$?

Sure, common sense says there's no solution. But, I feel, there should be one! (If there isn't, can't we construct one?)
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### What's the value of $i^i$? [duplicate]

What's the value of $i^i$?Is it real or imaginary?[$i$ here denotes imaginary number.]
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### Defining exponentiation on the integers

If one defines the integers as equivalence classes of pairs of natural numbers, there is a (canonical?) way to define addition and multiplication for the integers based on addition and multiplication ...
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### Solve $3^{2x} -2 \cdot 3^{x+5} + 3^{10} = 0$ for $x$

Here's the question: Solve for $x$ in $$3^{2x} - 2 \cdot 3^{x+5} + 3^{10} = 0$$ I know that you have to factor something out, I'm just not sure what that something is. Thanks in advance
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### Is there any notation for general $n$-th root $r$ such that $r^n=x$?

As we know that the notation for the $n$-th principal root is $\sqrt[n]{x}$ or $x^{1/n}$. But the principal root is not always the only possible root, e.g. for even $n$ and positive $x$ the principal ...
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### Which is the largest power of natural number that can be evaluated by computers? [closed]

Which is the largest power of natural number that can be evaluated by computers? For example if we take a very large power of 7: $7^{120000000000}$. Can a computer calculate this number?
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### How to find out the greater number from $15^{1/20}$ and $20^{1/15}$?

I have two numbers $15^{\frac{1}{20}}$ & $20^{\frac{1}{15}}$. How to find out the greater number out of above two? I am in 12th grade. Thanks for help!
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### How is N^2/3 equivalent to 1/(N^1/3)?

I've tried to look for similar things on StackExchange and elsewhere on the net, but can't seem to find anything, so thought I'd just ask for some help on here... Someone has kindly helped me with a ...
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### Converting a cryptographic hash to a string of English words: how many words are needed? (need help with exponentials)

A particular cryptographic hash is represented as a $57$ byte string, encoded as base $64$. RWSvUZXnw9gUb70PdeSNnpSmodCyIPJEGN1wWr+6Time1eP7KiWJ5eAM I want to ...
I tried mapping power functions onto the polar plane (i.e. converting x,y into r and $\theta$). I was successful with power functions representing $y=ax^n$ by $$r=\sqrt[n-1]{\frac ... 3answers 82 views ### x^x = y. Given y, find x. [duplicate] Title is fairly self-explanatory. For example, for y=27, x would be 3. Specifically I was trying to find x given y = 10^{100}, but I'd like to know how to solve it for any value of y. 4answers 144 views ### When does (x^x)^x=x^{(x^x)} in Real numbers? I have tried to solve this equation:(x^x)^x=x^{(x^x)} in real numbers I got only x=1,x=-1,x=2 , are there others solutions ? Note: x is real number . Thank you for your help . 2answers 49 views ### If a^{p}\cdot b^{p}= (a\cdot b)^{p} then why -1^{2}\cdot 3^{2}\neq (-1\cdot 3)^{2} If a^{p}\cdot b^{p}= (a\cdot b)^{p} then why$$-1^{2}\cdot 3^{2}\neq (-1\cdot 3)^{2}\\ -1\cdot 9\neq (-3)^{2}\\ -9\neq 9 I'm sorry, I don't know how to put latex code.
For example, $(a + b + c)^3 = a^3 + b^3 + c^3 + 3ab^2 + 3ac^2 + 3a^2b + 3a^2c + 3bc^2 + 3b^2c + 6abc$ Knowing the value of a, b and c, is there a way to find this without the coefficients i.e. \$a^3 + ...