What is the value of \sqrt{2^6 + 2^6 + 2^6 + 2^6}? — GMAT

What is the value of $\sqrt{2^6 + 2^6 + 2^6 + 2^6}$ ?

The four identical terms

2^6 + 2^6 + 2^6 + 2^6

sum to

4 \times 2^6

. Since

4 = 2^2

, the expression inside the radical becomes

2^2 \times 2^6 = 2^8

. Taking the square root yields

\sqrt{2^8} = 2^4 = 16

Express the sum of the four identical terms as a product.

2^6 + 2^6 + 2^6 + 2^6 = 4 \times 2^6

Adding a number to itself 4 times is equivalent to multiplying that number by 4.

Convert 4 to a power of 2 and combine exponents.

4 \times 2^6 = 2^2 \times 2^6 = 2^{2+6} = 2^8

When multiplying powers with the same base, add their exponents.

Simplify the square root of

2^8

\sqrt{2^8} = (2^8)^{1/2} = 2^4 = 16

Taking the square root of a power means dividing the exponent by 2.

Factoring identical exponential terms and applying exponent laws to simplify radicals.

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