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

1 Answer
Aug 11, 2016

#-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#