Simplify the expression. Write the result using positive exponents only. Assume that all bases are not equal to 0.
\[
\frac{\left(a^{5} b^{-9}\right)^{-5}}{\left(7 a^{2} b^{-1}\right)^{-2}}
\]
Answer
Rewrite a^-21 as 1/a^21. The final simplified expression is: \[\boxed{7^{2} \frac{b^{43}}{a^{21}}}\]
Step 1 ：Apply the power of a power rule to both the numerator and the denominator. The expression becomes: \[\frac{a^{-25} b^{45}}{7^{-2} a^{-4} b^{2}}\]
Step 2 ：Apply the rule of division of exponents to both a and b terms separately. The expression becomes: \[\frac{1}{7^{-2}} a^{-21} b^{43}\]
Step 3 ：Simplify the fraction 1/7^-2 to 7^2. The expression becomes: \[7^{2} a^{-21} b^{43}\]
Step 4 ：Rewrite a^-21 as 1/a^21. The final simplified expression is: \[\boxed{7^{2} \frac{b^{43}}{a^{21}}}\]
