# Linked Questions

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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 $x^\frac{2}{... 1answer 778 views ### Formal definition of numbers with real exponents [duplicate] The definition of the exponential with integer exponents is straightforward to define:$x^n=\underbrace{x\cdot\ldots\cdot x}_{n-\text{times}}$. These days I've been thinking about the formal ... 3answers 66 views ### How to define$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? 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 ... 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 ... 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 ... 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?... 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 ... 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 ... 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 ... 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: $$b^x = e^{\ln (b) x}$$ ... 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$\log(x)$: $$\log(x)... 1answer 88 views ### What does (-2)^x really mean? I think we can all agree that (-2)^{-1}=-1/2,(-2)^0=1,(-2)^1=-2,(-2)^2=4 But what does the function f=(-2)^x really mean? It is defined on the integers based on how most people understand ... 2answers 80 views ### “Complex” Roots, when can we compute result wihout imaginay numbers? [closed] The question is, how can we know if we can solve a n-index root without imaginary numbers when n-index is a non integer number, but a real number? First, basic theory:$$\sqrt[2.0]{8} = 2.8284.....$$... 1answer 61 views ### The concept of an real irrational power [duplicate] One can understand the concept of natural power, as x^n, being a product of a number by itself n times:$$x^n=\underbrace{x\cdot x\cdot\dotsb\cdot x}_n$\$ We can also get the idea of a rational ...

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