Which of the following is equivalent to \sqrt{\frac{8^{10} + 4^{15}}{8^4 + 4^6}}? — GMAT

Which of the following is equivalent to $\sqrt{\frac{8^{10} + 4^{15}}{8^4 + 4^6}}$ ?

The expression is equivalent to

2^9

By converting all terms to base 2,

8^{10} + 4^{15} = (2^3)^{10} + (2^2)^{15} = 2^{30} + 2^{30} = 2^{31}

in the numerator, and

8^4 + 4^6 = (2^3)^4 + (2^2)^6 = 2^{12} + 2^{12} = 2^{13}

in the denominator. The expression simplifies to

\sqrt{\frac{2^{31}}{2^{13}}} = \sqrt{2^{18}} = 2^9

Convert all terms in the numerator and denominator to a common base of 2.

Numerator:

8^{10} + 4^{15} = (2^3)^{10} + (2^2)^{15} = 2^{30} + 2^{30}

. Denominator:

8^4 + 4^6 = (2^3)^4 + (2^2)^6 = 2^{12} + 2^{12}

Expressing terms with prime bases enables simplification using exponent rules.

Factor out common terms to simplify addition in numerator and denominator.

Numerator:

2^{30} + 2^{30} = 2 \cdot 2^{30} = 2^{31}

. Denominator:

2^{12} + 2^{12} = 2 \cdot 2^{12} = 2^{13}

Adding two equal quantities

x + x

equals

2x

, increasing the power of 2 by 1.

Simplify the fraction inside the square root.

\frac{2^{31}}{2^{13}} = 2^{31 - 13} = 2^{18}.

Apply the quotient rule of exponents:

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

Evaluate the radical expression.

\sqrt{2^{18}} = (2^{18})^{1/2} = 2^{18/2} = 2^9.

Taking the square root of a power halves its exponent.

Exponents, Roots, and Powers of Integers

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