Use Multiplication Distributive Property: \({(xy)}^{a}={x}^{a}{y}^{a}\).
\[\frac{{5}^{2}{({a}^{2})}^{2}{b}^{2}\times -100{b}^{3}}{{({5}^{2}b)}^{2}}\]
Simplify \({5}^{2}\) to \(25\).
\[\frac{25{({a}^{2})}^{2}{b}^{2}\times -100{b}^{3}}{{(25b)}^{2}}\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{25{a}^{4}{b}^{2}\times -100{b}^{3}}{{(25b)}^{2}}\]
Take out the constants.
\[\frac{(25\times -100){a}^{4}{b}^{2}{b}^{3}}{{(25b)}^{2}}\]
Simplify \(25\times -100\) to \(-2500\).
\[\frac{-2500{a}^{4}{b}^{2}{b}^{3}}{{(25b)}^{2}}\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[\frac{-2500{a}^{4}{b}^{2+3}}{{(25b)}^{2}}\]
Simplify \(2+3\) to \(5\).
\[\frac{-2500{a}^{4}{b}^{5}}{{(25b)}^{2}}\]
Use Multiplication Distributive Property: \({(xy)}^{a}={x}^{a}{y}^{a}\).
\[\frac{-2500{a}^{4}{b}^{5}}{{25}^{2}{b}^{2}}\]
Simplify \({25}^{2}\) to \(625\).
\[\frac{-2500{a}^{4}{b}^{5}}{625{b}^{2}}\]
Move the negative sign to the left.
\[-\frac{2500{a}^{4}{b}^{5}}{625{b}^{2}}\]
Take out the constants.
\[-\frac{2500}{625}\times \frac{{a}^{4}{b}^{5}}{{b}^{2}}\]
Simplify \(\frac{2500}{625}\) to \(4\).
\[-4\times \frac{{a}^{4}{b}^{5}}{{b}^{2}}\]
Use Quotient Rule: \(\frac{{x}^{a}}{{x}^{b}}={x}^{a-b}\).
\[-4{a}^{4}{b}^{5-2}\]
Simplify \(5-2\) to \(3\).
\[-4{a}^{4}{b}^{3}\]
-4*a^4*b^3
