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(((1)/(9))^(124))/(((1)/(3))^(72)) Which of the following values is equal to the given value? Choose 1 answer: (A) 3^(-176) (B) 3^(-52) (C) 3^(52) (D) 3^(176)

(19)124(13)72 \frac{\left(\frac{1}{9}\right)^{124}}{\left(\frac{1}{3}\right)^{72}} \newlineWhich of the following values is equal to the given value?\newlineChoose 11 answer:\newline(A) 3−176 3^{-176} \newline(B) 3−52 3^{-52} \newline(C) 352 3^{52} \newline(D) 3176 3^{176}

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Full solution

Q. (19)124(13)72 \frac{\left(\frac{1}{9}\right)^{124}}{\left(\frac{1}{3}\right)^{72}} \newlineWhich of the following values is equal to the given value?\newlineChoose 11 answer:\newline(A) 3−176 3^{-176} \newline(B) 3−52 3^{-52} \newline(C) 352 3^{52} \newline(D) 3176 3^{176}
  1. Simplify expression using exponents: Simplify the given expression using the properties of exponents.\newlineThe given expression is (19)124/(13)72\left(\frac{1}{9}\right)^{124}/\left(\frac{1}{3}\right)^{72}. We know that (19)=3−2\left(\frac{1}{9}\right) = 3^{-2} and (13)=3−1\left(\frac{1}{3}\right) = 3^{-1}. Let's rewrite the expression using these equivalences.\newline(19)124/(13)72=(3−2)124/(3−1)72\left(\frac{1}{9}\right)^{124}/\left(\frac{1}{3}\right)^{72} = \left(3^{-2}\right)^{124} / \left(3^{-1}\right)^{72}\newlineNow, apply the power of a power rule (am)n=am∗n(a^{m})^{n} = a^{m*n}.\newline(3−2)124=3−2∗124=3−248\left(3^{-2}\right)^{124} = 3^{-2*124} = 3^{-248}\newline(3−1)72=3−1∗72=3−72\left(3^{-1}\right)^{72} = 3^{-1*72} = 3^{-72}\newlineSo the expression becomes 3−248/3−723^{-248} / 3^{-72}.
  2. Apply power of a power rule: Apply the quotient of powers rule to simplify the expression further.\newlineThe quotient of powers rule states that am/an=am−na^{m} / a^{n} = a^{m-n} when a≠0a \neq 0.\newline3−248/3−72=3−248−(−72)=3−248+72=3−1763^{-248} / 3^{-72} = 3^{-248 - (-72)} = 3^{-248 + 72} = 3^{-176}.
  3. Apply quotient of powers rule: Match the simplified expression to one of the given choices.\newlineThe simplified expression is 3−1763^{-176}, which corresponds to choice (A) 3−1763^{-176}.

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