Linked Questions

7
votes
5answers
639 views

What does $x^\pi$ mean? [duplicate]

I was just wondering, what does $x^\pi$ or for that matter, $x$ raised to any irrational number mean? For example, I want to represent $x^2$ then that would mean $x * x$ or if I want to do $x^\frac{2}{...
1
vote
1answer
778 views

Formal definition of numbers with real exponents [duplicate]

The definition of the exponential with integer exponents is straightforward to define: $x^n=\underbrace{x\cdot\ldots\cdot x}_{n-\text{times}}$. These days I've been thinking about the formal ...
0
votes
3answers
66 views

How to define $a^b$ when $b$ is real? [duplicate]

In my understanding, $64^{\frac23}$ can be defined as $(64^{\frac13})^2=4^2=16.$ How can I define $64^x$ when $x$ is real?
1
vote
1answer
113 views

What does $2^{2.5}$ mean [duplicate]

We know that $2^2$ means $2\cdot 2$, $2^3$ means $2\cdot 2\cdot 2$, and $2^5$ means $2\cdot 2\cdot 2\cdot 2\cdot 2$ and so on and so forth. But what does $2^{2.5}$ mean? I don't mean to say "How to ...
61
votes
11answers
6k views

What Is Exponentiation?

Is there an intuitive definition of exponentiation? In elementary school, we learned that $$ a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times}) $$ where $b$ is an integer. Then later on ...
3
votes
8answers
577 views

Do matrices have a “to the power of” operator?

Well I was sure that saying "$A^3$" (where $A$ is an $n\times n$ matrix) is nonsense. Sure one could do $(A\cdot A) A$ But that contains different operators etc. So what did my prof mean by the ...
1
vote
2answers
152 views

How would I compute a fractional exponent, such as $\bigl(\frac{9}{16}\bigr)^{3/2}$?

How would I determine $\left(\frac{9}{16}\right)^{\frac{3}{2}}$? I've never encountered an exponent that was a fraction, or a least never without a calculator. What steps would I take to simplify this?...
3
votes
2answers
89 views

How do we 'know' that $2^x$ is continuous?

It is intuitive for $2^n$, if $n$ is an integer, to exist. How do we know that less intuitive values such as $2^\frac{1}{2}$, $2^\sqrt{2}$, $2^\pi$ etc exist? I'd like to accept that $2^x$ is ...
3
votes
3answers
233 views

Why do real powers need positive bases? (and to handle them)

I'm studying Calculus and I've stumbled across a concept I have some difficulties in fully grasping. That is "real powers". I don't understand the theory behind it and I think I don't understand very ...
1
vote
1answer
111 views

Why is $\lim\limits_{x\to a} e^x = e^{\lim\limits_ {x\to a} x}$?

This is a confusion that I have had for a long time. " Why is $\lim\limits_{x\to a} e^x = e^{\lim\limits_ {x\to a} x}$?" Is there any proof or logic behind? Please explain. I have googled this and I ...
1
vote
2answers
75 views

How does exponentiation relate to multiplication?

My book derives the logarithm function as a definite integral of $1/x$ and defines the exponential function as its inverse. It then extends this definition to other bases: $$b^x = e^{\ln (b) x}$$ ...
1
vote
3answers
94 views

Avoiding circularity in explaining the meaning of real exponents

In an answer to 'What does $2^x$ really mean when $x$ is not an integer?' Álvaro Lozano-Robledo explains that we can understand real number exponents in terms of the definition of $\log(x)$: $$\log(x)...
1
vote
1answer
88 views

What does $(-2)^x$ really mean?

I think we can all agree that $(-2)^{-1}=-1/2,(-2)^0=1,(-2)^1=-2,(-2)^2=4$ But what does the function $f=(-2)^x$ really mean? It is defined on the integers based on how most people understand ...
2
votes
2answers
80 views

“Complex” Roots, when can we compute result wihout imaginay numbers? [closed]

The question is, how can we know if we can solve a n-index root without imaginary numbers when n-index is a non integer number, but a real number? First, basic theory: $$\sqrt[2.0]{8} = 2.8284.....$$...
1
vote
1answer
61 views

The concept of an real irrational power [duplicate]

One can understand the concept of natural power, as $x^n$, being a product of a number by itself $n$ times: $$x^n=\underbrace{x\cdot x\cdot\dotsb\cdot x}_n$$ We can also get the idea of a rational ...

15 30 50 per page