(-64)^3/2, simplify

Question

Recently I came across the question of negative 64 to the power of 3/2. I know how to simplify a positive number to a mixed number exponent, but not a negative number.

Answer 1

The square root of a negative number is not a real number. There are NO real numbers that multiply by themselves and get a negative answer, right? Any positive number multiplied by itself gives a positive. Any negative number multiplied by itself gives a positive.

But, we also have a whole category of numbers called imaginary numbers which DO include the square roots of negative numbers. The core of them is √-1 which we abbreviate using the letter "i".

So since √-1 = i and √-64 = √(64 • (-1)) we can say √-64 = √64 • i = 8i

We can also say that i² = -1

That means (-64)^3/2 = (8i)³ = 512i³ = 512i²•i = 512(-1)i = -512i