# Tagged Questions

56 views

### negative powers $(x^{-2} = 1/x^2)$

I need clarification for negative power of a number. I understand $x$ to the power of $2$ is equal to $x\cdot x$ But how $x$ to the power of $-2$ is equal to $\dfrac{1}{x^2}$ ?
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### Will negative bases with irrational exponents get a real or imaginary number?

Here are a few examples: $$(-1)^{\sqrt{2}},(-2)^{\pi},(-3)^{e}$$ From what I've learned, negative bases must have denominators of the exponent odd. Normally if we do $(-2)^{0.258}$ it would be the ...
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### The shape of a graph of a function with $n$th-roots?

Not just these type of functions: $$\sqrt[3]{x}=x^{1/3} \;\;\;\text{and} \;\;\; \sqrt[8]{x}=x^{1/8}$$ But also more complicated expressions, like expressions that have $n$th roots inside of ...
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### Is the fraction of the irrational exponentiations of two coprime integers by a rational an irrational?

Consider two strictly positive integer coprimes $n, m\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. Consider furthermore that the three number statifies the following condition: ...
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### Can the exponentiation of an integer by a rational be a non-integer rational?

Consider a strictly positive integer $n\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. My question is the following: what is the nature of $n^r$? My first guess is that $n^r$ is an ...
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### Fraction raised to integer power

if I have $(p/q)^n$ where $p,q,n$ are integers and $p/q$ is a... I don't know what you call it. Not a whole number, but something like 15/7 where you can't reduce it any more and it's non-integer. Can ...
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### Rational number to the power of irrational number = irrational number. True?

I suggested the following problem to my friend: prove that there exist irrational numbers $a$ and $b$ such that $a^b$ is rational. The problem seems to have been discussed in this question. Now, his ...
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### Any irrational number can be raised to a power so that the result is an integer number [duplicate]

Does it hold in general, that for every irrational number there exists a power to which when raised, the result will be an integer? Does there exist a counterexample, of which it can be showed that no ...
237 views

### A calculator's solution to irrational exponent

An irrational number cannot be represented by $\frac{p}{q}$ where $p$ and $q$ are integers. And when we encounter exponents with decimal points, it is a possible way and a rather simple one to turn ...
331 views

### Is it possible to prove the positive root of the equation ${^4}x=2$, $x=1.4466014324…$ is irrational?

(somewhat related to my earlier question) Let ${^n}a$ denote tetration $\underbrace{a^{a^{.^{.^{.^a}}}}}_{n \text{ times}}$ (or, defined recursively, ${^1}a=a$, ${^{n+1}}a=a^{({^n}a)}$). The ...
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### Can you raise a Matrix to a non integer number? [duplicate]

So I heard you can take a matrix A to the power 2, take it to a -3th power and multiply it by an irrational number. You can also do some other non-intuitive things like taking e to the power of a ...
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### Extending the rationals using exponentiation

The set of integers can be constructed as an equivalence relation over the natural numbers using the the binary operation of addition, and a similar process yields the rationals from integers and ...
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### Exponentiation when the exponent is irrational

I am just curious about what inference we can draw when we calculate something like $$\text{base}^\text{exponent}$$ where base = rational or irrational number and exponent = irrational number
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### What is the exact definition of a rational power?

I was taught in school that $$x^{a/b} = \sqrt[b]{x^a}$$ however, wolfram says this is not always true: $\sqrt[3]{x^2} \ne x^{2/3}$ ...
5k views

### Can you raise a number to an irrational exponent?

The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. ...
Suppose I show that: $$x^{f(z)/g(z)} = y \pmod{4}$$ is impossible for some given positive integers $x$ and $y$, where, \begin{align*} f(z) &= \phi(4) k_1(z) + 1 \\ &= 2 k_1(z) + 1\\ g(z) ...