What is the value of the expression \frac{\sqrt{144} \times 2^{-3}}{3^2}? — ACT

What is the value of the expression $\frac{\sqrt{144} \times 2^{-3}}{3^2}$ ?

The correct answer is the option representing

\frac{1}{6}

The correct answer is obtained by evaluating each part of the expression: the square root

\sqrt{144}

12

, the negative exponent

2^{-3}

\frac{1}{8}

, and the denominator

3^2

9

. Multiplying the numerator terms yields

12 \times \frac{1}{8} = 1.5

, and dividing by the denominator

9

gives

\frac{1.5}{9} = \frac{1}{6}

Simplify the square root term in the numerator,

\sqrt{144}

\sqrt{144} = 12

Since

12^2 = 144

, the principal square root of

144

12

Simplify the negative exponent term in the numerator,

2^{-3}

2^{-3} = \frac{1}{2^3} = \frac{1}{8}

A negative exponent indicates the reciprocal of the base raised to the positive power,

a^{-n} = \frac{1}{a^n}

Calculate the value of the numerator by multiplying the simplified terms.

Numerator

= 12 \times \frac{1}{8} = \frac{12}{8} = \frac{3}{2}

Multiply the two simplified parts of the numerator.

Simplify the denominator,

3^2

3^2 = 9

3^2

means

3 \times 3

, which equals

9

Divide the simplified numerator by the simplified denominator to find the final value.

\frac{3/2}{9} = \frac{3}{2 \times 9} = \frac{3}{18} = \frac{1}{6}

Divide fractions by multiplying by the reciprocal of the denominator.

Evaluating expressions involving square roots, negative integer exponents, and positive integer exponents.

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