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(a^(m))^(n)*a^(mn) Which of the following is equivalent to the given expression? Choose 1 answer: (A) 2a^(mn) (B) a^(2mn) (C) a^(m^(2)n^(2)) (D) a^(m^(n)+mn)

(am)n⋅amn \left(a^{m}\right)^{n} \cdot a^{m n} \newlineWhich of the following is equivalent to the given expression?\newlineChoose 11 answer:\newline(A) 2amn 2 a^{m n} \newline(B) a2mn a^{2 m n} \newline(C) am2n2 a^{m^{2} n^{2}} \newline(D) amn+mn a^{m^{n}+m n}

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Full solution

Q. (am)n⋅amn \left(a^{m}\right)^{n} \cdot a^{m n} \newlineWhich of the following is equivalent to the given expression?\newlineChoose 11 answer:\newline(A) 2amn 2 a^{m n} \newline(B) a2mn a^{2 m n} \newline(C) am2n2 a^{m^{2} n^{2}} \newline(D) amn+mn a^{m^{n}+m n}
  1. Apply Power Rule: Use the power of a power rule, which states that (am)n=am∗n(a^m)^n = a^{m*n}.\newline(a(m))n=am∗n(a^{(m)})^n = a^{m*n}
  2. Use Product Rule: Now, multiply the result by amna^{mn} using the product of powers rule, which states that ab⋅ac=ab+ca^b \cdot a^c = a^{b+c}. \newlineam⋅n⋅amn=am⋅n+mna^{m\cdot n} \cdot a^{mn} = a^{m\cdot n + mn}
  3. Factor out Common Factor: Simplify the exponent by factoring out the common factor mm. am∗n+mn=am(n+n)a^{m*n + mn} = a^{m(n + n)}
  4. Simplify Exponent: Since n+n=2nn + n = 2n, the expression simplifies further.\newlineam(n+n)=am⋅2na^{m(n + n)} = a^{m\cdot 2n}
  5. Further Simplification: Finally, we have the simplified expression. a(m∗2n)=a2mna^{(m*2n)} = a^{2mn}

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