If x is a real number such that x^4 = 16, which of the following values could be equal to … — GRE General Test

If $x$ is a real number such that $x^4 = 16$ , which of the following values could be equal to $x^3$ ? Select all that apply.

The values

-8

and

8

are the possible values of

x^3

Solving

x^4 = 16

gives two real solutions,

x = 2

and

x = -2

, because raising any real number to an even power yields a non-negative result. Cubing each solution gives

2^3 = 8

and

(-2)^3 = -8

. Thus, both

-8

and

8

are correct values for

x^3

Find all real solutions for

x

in the equation

x^4 = 16

x = 2

x = -2

Taking the fourth root of both sides gives

|x| = \sqrt[4]{16} = 2

, so

x

can be positive or negative.

Calculate

x^3

for the positive root

x = 2

2^3 = 8

Cubing a positive number yields a positive result.

Calculate

x^3

for the negative root

x = -2

(-2)^3 = -8

Cubing a negative number yields a negative result because an odd exponent preserves the sign.

Even and odd power rules for positive and negative real bases

Question