If x is a negative real number, which of the following expressions are equivalent to \sqrt… — GRE General Test

If $x$ is a negative real number, which of the following expressions are equivalent to $\sqrt{-x^3}$ ? Select all such expressions.

The expressions equivalent to

\sqrt{-x^3}

are

-x\sqrt{-x}

|x|\sqrt{-x}

, and

(-x)^{3/2}

Because

x

is negative,

-x

is a positive quantity. We can express

-x^3

(-x)^3 = (-x)^2 \cdot (-x)

. Taking the principal square root yields

\sqrt{(-x)^2 \cdot (-x)} = \sqrt{(-x)^2} \cdot \sqrt{-x} = -x\sqrt{-x}

. Because

|x| = -x

for negative numbers, the expression

|x|\sqrt{-x}

is identical to

-x\sqrt{-x}

. Furthermore, converting to rational exponents gives

(-x)^{3/2} = \sqrt{(-x)^3} = \sqrt{-x^3}

. Thus, all three of these expressions are mathematically equivalent to the original radical expression.

Analyze the sign of the base and inside of the radical

Since

x < 0

, the quantity

-x

is strictly positive (

-x > 0

). Consequently,

-x^3 = (-x)^3 > 0

, ensuring

\sqrt{-x^3}

is a real, non-negative number.

Principal square roots require a non-negative radicand and yield a non-negative result in real arithmetic.

Simplify the radical expression using perfect squares

\sqrt{-x^3} = \sqrt{(-x)^2 \cdot (-x)} = \sqrt{(-x)^2} \cdot \sqrt{-x} = (-x)\sqrt{-x} = -x\sqrt{-x}

Because

-x > 0

\sqrt{(-x)^2} = -x

Evaluate equivalent representations using absolute value and rational exponents

Since

x < 0

|x| = -x

, so

|x|\sqrt{-x} = -x\sqrt{-x}

. Also,

(-x)^{3/2} = \sqrt{(-x)^3} = \sqrt{-x^3}

Both rewrite rules preserve both magnitude and non-negative sign for all

x < 0

Simplifying radicals and fractional exponents with negative variable bases

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