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2^(5)*((1)/(4))^(7) Which of the following expressions is equivalent to the given expression? Choose 1 answer: (A) 2^(-2) (B) 2^(-4) (c) 2^(-9) (D) 2^(10)

25⋅(14)7 2^{5} \cdot\left(\frac{1}{4}\right)^{7} \newlineWhich of the following expressions is equivalent to the given expression?\newlineChoose 11 answer:\newline(A) 2−2 2^{-2} \newline(B) 2−4 2^{-4} \newline(C) 2−9 2^{-9} \newline(D) 210 2^{10}

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Full solution

Q. 25⋅(14)7 2^{5} \cdot\left(\frac{1}{4}\right)^{7} \newlineWhich of the following expressions is equivalent to the given expression?\newlineChoose 11 answer:\newline(A) 2−2 2^{-2} \newline(B) 2−4 2^{-4} \newline(C) 2−9 2^{-9} \newline(D) 210 2^{10}
  1. Understand Expression: Understand the given expression and the properties of exponents.\newlineWe have the expression 25×(14)72^{5}\times\left(\frac{1}{4}\right)^{7}. We know that (14)\left(\frac{1}{4}\right) is the same as 2−22^{-2} because 14=222=2122=2(1−2)=2−1\frac{1}{4} = \frac{2}{2^2} = \frac{2^1}{2^2} = 2^{(1-2)} = 2^{-1}.
  2. Rewrite Using Property: Rewrite the expression using the property from Step 11.\newlineWe can rewrite (14)7\left(\frac{1}{4}\right)^{7} as (2−2)7(2^{-2})^{7}.\newlineSo, the expression becomes 25⋅(2−2)7.2^{5} \cdot (2^{-2})^{7}.
  3. Apply Power Rule: Apply the power of a power rule to the second term.\newlineUsing the power of a power rule (am)n=am∗n(a^{m})^{n} = a^{m*n}, we get:\newline(2−2)7=2−2∗7=2−14(2^{-2})^{7} = 2^{-2*7} = 2^{-14}.\newlineNow, the expression is 25×2−142^{5} \times 2^{-14}.
  4. Combine Terms: Apply the product of powers rule to combine the terms.\newlineUsing the product of powers rule am×an=a(m+n)a^m \times a^n = a^{(m+n)}, we combine the terms:\newline25×2−14=2(5−14)=2−9.2^{5} \times 2^{-14} = 2^{(5-14)} = 2^{-9}.
  5. Match with Options: Match the result with the given options.\newlineThe simplified expression is 2−92^{-9}, which matches option (C)(C).

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