If x = 4, what is the value of \sqrt{9^x + 9^x + 9^x + 9^x}? — GRE General Test

If $x = 4$ , what is the value of $\sqrt{9^x + 9^x + 9^x + 9^x}$ ?

162

Combining the four identical terms under the square root gives

\sqrt{4 \cdot 9^4}

. Splitting the root using product rules yields

\sqrt{4} \cdot \sqrt{9^4} = 2 \cdot 9^2 = 2 \cdot 81 = 162

Substitute

x = 4

into the given radical expression.

The expression becomes

\sqrt{9^4 + 9^4 + 9^4 + 9^4}

Direct substitution of the given variable value.

Combine the four identical terms under the radical.

9^4 + 9^4 + 9^4 + 9^4 = 4 \cdot 9^4

Repeated addition of

4

identical terms is equivalent to multiplication by

4

Apply the product rule for square roots,

\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}

\sqrt{4 \cdot 9^4} = \sqrt{4} \cdot \sqrt{9^4} = 2 \cdot 9^2.

Both

4

and

9^4

are perfect squares.

Evaluate the numerical value.

2 \cdot 81 = 162.

Simplifying the arithmetic expression.

Combining like terms under radical sign and applying product rules of square roots

Estimated Time:1m 15s

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