Simplify (3AB)^-1(A^2B^-1)^2

Problem

(3*A*B)^(−1)*(A^2*B^(−1))^2

Solution

1. Apply the power of a product rule to the first term (3*A*B)^(−1) by distributing the exponent −1 to each factor inside the parentheses.

(3*A*B)^(−1)=3^(−1)*A^(−1)*B^(−1)

2. Apply the power of a power rule to the second term (A^2*B^(−1))^2 by multiplying the exponents.

(A^2*B^(−1))^2=A^(2⋅2)*B^(−1⋅2)

(A^2*B^(−1))^2=A^4*B^(−2)

3. Combine the terms by multiplying the results from the previous steps.

3^(−1)*A^(−1)*B^(−1)*A^4*B^(−2)

4. Group like bases and add their exponents according to the product rule x^m*x^n=x^(m+n)

3^(−1)*A^(−1+4)*B^(−1−2)

3^(−1)*A^3*B^(−3)

5. Rewrite with positive exponents by moving terms with negative exponents to the denominator.

(A^3)/(3^1*B^3)

(A^3)/(3*B^3)

Final Answer

(3*A*B)^(−1)*(A^2*B^(−1))^2=(A^3)/(3*B^3)

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