What is the value of \sqrt{16^5 + 16^5 + 16^5 + 16^5}? — GMAT

What is the value of $\sqrt{16^5 + 16^5 + 16^5 + 16^5}$ ?

2^{11}

Combining the four identical terms inside the radical gives

4 \cdot 16^5

. Expressing both factors with base 2 yields

2^2 \cdot (2^4)^5 = 2^2 \cdot 2^{20} = 2^{22}

. Taking the square root gives

\sqrt{2^{22}} = 2^{11}

, which matches the value

2^{11}

Factor out the common term inside the square root.

16^5 + 16^5 + 16^5 + 16^5 = 4 \cdot 16^5

Adding four identical terms is equivalent to multiplying the term by 4.

Convert both numbers to powers of 2.

4 = 2^2

and

16^5 = (2^4)^5 = 2^{20}

, so

4 \cdot 16^5 = 2^2 \cdot 2^{20} = 2^{22}

Expressing terms with a common prime base allows exponent rules to be applied.

Apply the square root to the simplified exponential expression.

\sqrt{2^{22}} = (2^{22})^{1/2} = 2^{11}

Taking the square root of a base raised to a power is equivalent to dividing the exponent by 2.

Combining like terms with exponents and applying prime factorization exponent laws under radical signs.

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