If x = \frac{\sqrt{8^{12} \times 4^6}}{\sqrt[3]{16^9}}, what is the value of \sqrt{x}? — ACT

If $x = \frac{\sqrt{8^{12} \times 4^6}}{\sqrt[3]{16^9}}$ , what is the value of $\sqrt{x}$ ?

A
16
64Answer
C
256
D
4,096
E
65,536

Answer

First, convert all bases in the expression to base 2:

8^{12} = (2^3)^{12} = 2^{36}

4^6 = (2^2)^6 = 2^{12}

, and

16^9 = (2^4)^9 = 2^{36}

. The numerator becomes

\sqrt{2^{36} \times 2^{12}} = \sqrt{2^{48}} = 2^{24}

. The denominator becomes

\sqrt[3]{2^{36}} = 2^{12}

. Dividing the numerator by the denominator gives

x = \frac{2^{24}}{2^{12}} = 2^{12}

. Finally, taking the square root of

x

yields

\sqrt{x} = \sqrt{2^{12}} = 2^6 = 64

Step-by-Step Solution

Express each base in the expression (

8

4

, and

16

) as a power of

2

The bases are rewritten as

8 = 2^3

4 = 2^2

, and

16 = 2^4

Converting all terms to a common base allows us to apply standard laws of exponents to simplify the expression.

Simplify the numerator

\sqrt{8^{12} \times 4^6}

using exponent rules.

\sqrt{(2^3)^{12} \times (2^2)^6} = \sqrt{2^{36} \times 2^{12}} = \sqrt{2^{48}} = 2^{24}

We multiply the exponents for the power of a power rule, add the exponents when multiplying powers with the same base, and then divide the exponent by 2 to take the square root.

Simplify the denominator

\sqrt[3]{16^9}

using exponent rules.

\sqrt[3]{(2^4)^9} = \sqrt[3]{2^{36}} = 2^{12}

We multiply the exponents for the power of a power rule and then divide the exponent by 3 to take the cube root.

Evaluate

x

by dividing the simplified numerator by the simplified denominator.

x = \frac{2^{24}}{2^{12}} = 2^{24 - 12} = 2^{12}

We subtract the exponent of the denominator from the exponent of the numerator when dividing powers with the same base.

Calculate the value of

\sqrt{x}

as requested in the question.

\sqrt{x} = \sqrt{2^{12}} = 2^{12/2} = 2^6 = 64

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

Key Concept

Applying properties of exponents and radical operations to simplify rational expressions.

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