Question

Let x be a nonnegative real number. Determine whether the statement is always, sometimes, or never true. Justify your answer.

x^(1/3) = x^3
a) This statement is sometimes true; it's false when x= 0 or x=1, but otherwise true.
b) This statement is sometimes true; it's true when x=0, x=1, or x= -1, but otherwise false.
c) This statement is never true because of the Definition of Rational Exponents.
d) This statement is always true because of the Definition of Rational Exponents.

Answer 1

The equation x^(1/3) = x^3 is sometimes true specifically when x equals 0 or 1. For all other nonnegative real numbers, the equation does not hold because the outcomes of a cube root and cubing are not the same.

Step-by-step explanation:

The statement x^(1/3) = x^3 is concerning the equality of two different powers of a nonnegative real number x. To determine whether this equality holds true, we consider what each side represents. The left side, x^(1/3), is asking for the cube root of x, while the right side, x^3, is asking for x multiplied by itself three times. These two expressions are not generally equivalent. For any x other than 0 and 1, raising x to the 1/3 power will not result in the same value as raising x to the 3rd power. There are only two cases where the equation holds: when x = 0, and when x = 1. If x = 0, then both sides of the equation are 0, since any nonnegative power of 0 is 0. If x = 1, then both sides of the equation are 1, since any power of 1 is 1. Therefore, the correct answer to the question is: Option b: This statement is sometimes true; it's true when x=0, x=1, but otherwise false.