How do you solve root3(x^2)= 9?

2 Answers
Mar 30, 2017

Undo each of the things done to x one at a time.

Explanation:

First, to undo the cube root, we can cube both sides to get

x^2=729

Next, to undo the square, we can take the square root of both sides to get

x=+-27

Mar 30, 2017

x=+-27

Explanation:

Given:

root(3)(x^2)=9

Note that 9=3^2, so this can also be written:

root(3)(x^2) = 3^2

Note that both t rarr t^3 and its inverse t rarr root(3)(t) are one to one as functions of real numbers. So if we cube both sides of the equation then we neither lose any solution nor introduce any extraneous solutions:

Cubing both sides of the equation, we get:

x^2 = (3^2)^3 = 3^(2*3) = 3^(3*2) = (3^3)^2 = 27^2

Subtract 27^2 from both ends to get:

x^2-27^2 = 0

The difference of squares identity can be written:

a^2-b^2 = (a-b)(a+b)

Using this with a=x and b=27 we find:

0 = x^2-27^2 = (x-27)(x+27)

So:

x = +-27