+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.
+118703 (-1)^1.5
(−1)1.5=(−1)32I can look at this in 2 different ways so lets see what happens.=[(−1)3]12or[(−1)12]3=[−1]12or[√−1]3=√−1or[i]3=±ior−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)12⇒√−1 is a law of fractional exponents. In general terms, xab=b√xa=(b√x)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.
x3∗x2=x3+2=x5
Let's try another example but with fractional exponents.
312∗312=312+12=31=3
In this example here, 312 is a number that when multiplied by itself yields 3. That sounds like the definition of the square root, doesn't it? Therefore, 312=√3. One can also expound upon this.
| 313∗313∗313=313+13+13=31=3 |
| 314∗314∗314∗314=314+14+14+14=31=3 |
| 315∗315∗315∗315∗315=315+15+15+15+15=31=3 |
| 31x∗31x∗31x∗...∗31x∗31x=31x+1x+1x+...+1x+1x=31=3 |
The pattern continues. 313 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. 31x, when multiplied x times, yields 3.
