If x is a positive real number, which of the following is equivalent to the expression \fr… — GRE General Test

If $x$ is a positive real number, which of the following is equivalent to the expression $\frac{(x^3)^2 \cdot x^{-4}}{\sqrt{x^8}}$ ?

The simplified expression is equivalent to

x^{-2}

Applying the rules of exponents systematically yields

(x^3)^2 = x^6

in the numerator, which combines with

x^{-4}

to give

x^2

. The denominator

\sqrt{x^8}

simplifies to

x^{8/2} = x^4

. Dividing

x^2

x^4

gives

x^{2-4} = x^{-2}

Simplify the power raised to a power in the numerator.

(x^3)^2 = x^{3 \cdot 2} = x^6

When raising a power to a power, multiply the exponents:

(a^m)^n = a^{m \cdot n}

Multiply the terms in the numerator.

x^6 \cdot x^{-4} = x^{6 + (-4)} = x^2

When multiplying exponential expressions with the same base, add the exponents:

a^m \cdot a^n = a^{m+n}

Simplify the radical in the denominator.

\sqrt{x^8} = (x^8)^{\frac{1}{2}} = x^{\frac{8}{2}} = x^4

Taking the square root of a non-negative term is equivalent to raising it to the power of

\frac{1}{2}

Divide the numerator by the denominator.

\frac{x^2}{x^4} = x^{2 - 4} = x^{-2}

When dividing exponential expressions with the same base, subtract the denominator exponent from the numerator exponent: \frac{a^m}{a^n} = a^{m-n}.

Laws of Exponents and Radical Simplification

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