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$Which fraction has the same value as 6^(-2)*24^((1)/(2)) ? (A) (1)/(144) (B) (sqrt2)/(6) (C) (sqrt6)/(18) (D) (1)/(3)$

Which fraction has the same value as $6^{-2} \cdot 24^{\frac{1}{2}}$ ? $\newline$ (A) $\frac{1}{144}$ $\newline$ (B) $\frac{\sqrt{2}}{6}$ $\newline$ (C) $\frac{\sqrt{6}}{18}$ $\newline$ (D) $\frac{1}{3}$

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Multiplication with rational exponents

Full solution

Q. Which fraction has the same value as

6^{-2} \cdot 24^{\frac{1}{2}}

?

\newline

(A)

\frac{1}{144}

\newline

(B)

\frac{\sqrt{2}}{6}

\newline

(C)

\frac{\sqrt{6}}{18}

\newline

(D)

\frac{1}{3}

Evaluate Exponent: Evaluate $6^{-2}$ $\newline$ The negative exponent means that we take the reciprocal of the base raised to the positive exponent. $\newline$ $6^{-2} = \frac{1}{6^2} = \frac{1}{36}$
Find Square Root: Evaluate $24^{\left(\frac{1}{2}\right)}$ $\newline$ The exponent $\left(\frac{1}{2}\right)$ means that we take the square root of the base. $\newline$ $24^{\left(\frac{1}{2}\right)} = \sqrt{24} = \sqrt{4\cdot6} = \sqrt{4}\cdot\sqrt{6} = 2\cdot\sqrt{6}$
Multiply Results: Multiply the results from Step $1$ and Step $2$ $\newline$ Now we multiply $\frac{1}{36}$ by $2\sqrt{6}$ . $\newline$ $(\frac{1}{36}) \times (2\sqrt{6}) = \frac{2\sqrt{6}}{36}$
Simplify Fraction: Simplify the fraction $\newline$ We can simplify the fraction by dividing both the numerator and the denominator by $2$ . $\newline$ $\frac{2\sqrt{6}}{36} = \frac{\sqrt{6}}{18}$
Match Given Options: Match the result with the given options $\newline$ The simplified expression from Step $4$ is $\frac{\sqrt{6}}{18}$ , which matches option (C).

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