+12 Furthermore, how/why do powers of non-integers work? For example, why would (-1)^1.5 be -i?
+655 (-1) ^ 1.5 is 1.
2^0.5 is basically just multiplying 2 0.5 times, which is square root.
+118687 (-1)^1.5
\((-1)^{1.5}\\ = (-1)^{\frac{3}{2}}\\ \text{I can look at this in 2 different ways so lets see what happens.}\\ = [(-1)^3]^{\frac{1}{2}}\qquad or \qquad [(-1)^\frac{1}{2}]^{3}\\ = [-1]^{\frac{1}{2}}\qquad or \qquad [\sqrt{-1}]^{3}\\ =\sqrt{-1}\qquad or \qquad [i]^{3}\\ =\pm i \qquad \qquad or \qquad -i\\ \)
The only common answer is -i so I guess that is the only correct answer.
Maybe another mathematician would like to comment here......
+2446 \((-1)^{\frac{1}{2}}\Rightarrow\sqrt{-1}\) is a law of fractional exponents. In general terms, \(x^{\frac{a}{b}}=\sqrt[b]{x^a}=\left(\sqrt[b]{x}\right)^a\)
Why is this the case? Well, I can attempt to explain it to you.
The law of exponents explains how to handle multiplication of exponents with identical bases.
\(x^3*x^2=x^{3+2}=x^5\)
Let's try another example but with fractional exponents.
\(3^{\frac{1}{2}}*3^{\frac{1}{2}}=3^{\frac{1}{2}+\frac{1}{2}}=3^1=3\)
In this example here, \(3^{\frac{1}{2}}\) is a number that when multiplied by itself yields 3. That sounds like the definition of the square root, doesn't it? Therefore, \(3^{\frac{1}{2}}=\sqrt{3}\). One can also expound upon this.
| \(3^{\frac{1}{3}}*3^{\frac{1}{3}}*3^{\frac{1}{3}}=3^{{\frac{1}{3}}+{\frac{1}{3}}+{\frac{1}{3}}}=3^1=3\) |
| \(3^{\frac{1}{4}}*3^{\frac{1}{4}}*3^{\frac{1}{4}}*3^{\frac{1}{4}}=3^{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}=3^1=3\) |
| \(3^{\frac{1}{5}}*3^{\frac{1}{5}}*3^{\frac{1}{5}}*3^{\frac{1}{5}}*3^{\frac{1}{5}}=3^{\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}}=3^1=3\) |
| \(3^{\frac{1}{x}}*3^{\frac{1}{x}}*3^{\frac{1}{x}}*...*3^{\frac{1}{x}}*3^{\frac{1}{x}}=3^{\frac{1}{x}+\frac{1}{x}+\frac{1}{x}+...+\frac{1}{x}+\frac{1}{x}}=3^1=3\) |
The pattern continues. \(3^{\frac{1}{3}}\) is a number that when multiplied by itself 3 times yields 3. This is the definition of the cubic root. Then, as you can see, I made a generalization. \(3^{\frac{1}{x}}\), when multiplied x times, yields 3.
