Question

simplify the expression. assume that variables represent nonzero real numbers. (3a^(-2))^(2)(a^(7))^(-1)

Explanation:

Step1: Expand first term

Use power - of - a - product rule \((ab)^n=a^n b^n\) and power - of - a - power rule \((a^m)^n=a^{mn}\). \((3a^{-2})^2 = 3^2\times(a^{-2})^2=9a^{-4}\).

Step2: Expand second term

Use power - of - a - power rule on \((a^{7})^{-1}\), we get \(a^{-7}\).

Step3: Multiply the two terms

When multiplying terms with the same base \(a\), use the rule \(a^m\times a^n=a^{m + n}\). So \(9a^{-4}\times a^{-7}=9a^{-4+( - 7)}=9a^{-11}\).

Step4: Rewrite with positive exponent

Use the rule \(a^{-n}=\frac{1}{a^{n}}\), so \(9a^{-11}=\frac{9}{a^{11}}\).

Answer:

\(\frac{9}{a^{11}}\)