If a > 0, which of the following is equivalent to the expression \sqrt{\frac{a^8 \cdot a^4… — GRE General Test

If $a > 0$ , which of the following is equivalent to the expression $\sqrt{\frac{a^8 \cdot a^4}{a^{-2}}}$ ?

a^7

Multiplying the terms in the numerator gives

a^{12}

. Dividing by

a^{-2}

gives

a^{12 - (-2)} = a^{14}

. Taking the square root of

a^{14}

gives

(a^{14})^{1/2} = a^7

Simplify the numerator inside the square root using the product rule

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

a^8 \cdot a^4 = a^{8+4} = a^{12}

Powers with the same base are multiplied by adding their exponents.

Divide by the denominator using the quotient rule

\frac{a^m}{a^n} = a^{m-n}

\frac{a^{12}}{a^{-2}} = a^{12 - (-2)} = a^{14}

Dividing powers with the same base requires subtracting the lower exponent from the upper exponent.

Apply the fractional exponent rule for radicals

\sqrt{x} = x^{\frac{1}{2}}

\sqrt{a^{14}} = (a^{14})^{\frac{1}{2}} = a^{14 \cdot \frac{1}{2}} = a^7

Taking the square root of a power is equivalent to multiplying the exponent by

\frac{1}{2}

Simplifying expressions using exponent laws and fractional radical powers

Question