Question

How do you simplify `-8 ^ (2/3)`?

Answer 1

`-4`

Explanation:

Using the `color(blue)"laws of exponents"`

`color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(a^mxxa^n=a^(m+n))color(white)(a/a)|)))........ (A)`

Let's define, generally, the meaning of `a^(2/3)`

Using (A)

`a^(2/3)xxa^(2/3)xxa^(2/3)=a^(6/3)=a^2........ (1)`

now `a^(2/3)xxa^(2/3)xxa^(2/3)=(a^(2/3))^3........ (2)`

Since the 2 expressions are equivalent we can equate them.

`rArr(a^(2/3))^3=a^2`

Take the `color(blue)"cube root"` of both sides

`rArrcolor(red)(|bar(ul(color(white)(a/a)color(black)(a^(2/3)=root(3)(a^2)=(root(3)(a))^2)color(white)(a/a)|)))`

We can now evaluate `-8^(2/3)`

`-8^(2/3)=-1xx(root(3)8)^2=-1xx(2)^2=-1xx4=-4`