6.5 Divide monomials  (Page 3/4)

Solution

ⓐ

Use the Quotient Property, ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$ .
Simplify.

Use the Quotient Property, ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$ .
Simplify.

Raise the numerator and denominator to the third power.

Solution

$\begin{array}{cccc}& & & \hfill \frac{{\left(y^4\right)}^2}{y^6}\hfill \\ \text{Multiply the exponents in the numerator.}\hfill & & & \hfill \frac{y^8}{y^6}\hfill \\ \text{Subtract the exponents.}\hfill & & & \hfill y^2\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \hfill \frac{b^{12}}{{\left(b^2\right)}^6}\hfill \\ \text{Multiply the exponents in the denominator.}\hfill & & & \hfill \frac{b^{12}}{b^{12}}\hfill \\ \text{Subtract the exponents.}\hfill & & & \hfill b^0\hfill \\ \text{Simplify.}\hfill & & & \hfill 1\hfill \end{array}$

Notice that after we simplified the denominator in the first step, the numerator and the denominator were equal. So the final value is equal to 1.

Solution

$\begin{array}{cccc}& & & \hfill {\left(\frac{y^9}{y^4}\right)}^2\hfill \\ \begin{array}{c}\text{Remember parentheses come before exponents.}\hfill \\ \text{Notice the bases are the same, so we can simplify}\hfill \\ \text{inside the parentheses. Subtract the exponents.}\hfill \end{array}\hfill & & & \hfill {\left(y^5\right)}^2\hfill \\ \text{Multiply the exponents.}\hfill & & & \hfill y^{10}\hfill \end{array}$

Solution

Here we cannot simplify inside the parentheses first, since the bases are not the same.

$\begin{array}{cccc}& & & \hfill {\left(\frac{j^2}{k^3}\right)}^4\hfill \\ \begin{array}{c}\text{Raise the numerator and denominator to the third power}\hfill \\ \text{using the Quotient to a Power Property,}{\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}.\hfill \end{array}\hfill & & & \hfill \frac{{\left(j^2\right)}^4}{{\left(k^3\right)}^4}\hfill \\ \text{Use the Power Property and simplify.}\hfill & & & \hfill \frac{j^8}{k^{12}}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \hfill {\left(\frac{2m^2}{5n}\right)}^4\hfill \\ \begin{array}{c}\text{Raise the numerator and denominator to the fourth}\hfill \\ \text{power, using the Quotient to a Power Property,}{\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}.\hfill \end{array}\hfill & & & \hfill \frac{{(2m^2)}^4}{{(5n)}^4}\hfill \\ \text{Raise each factor to the fourth power.}\hfill & & & \hfill \frac{2^4{(m^2)}^4}{5^4{n}^4}\hfill \\ \text{Use the Power Property and simplify.}\hfill & & & \hfill \frac{16m^8}{625n^4}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \hfill \frac{{\left(x^3\right)}^4{\left(x^2\right)}^5}{{\left(x^6\right)}^5}\hfill \\ \text{Use the Power Property,}{\left(a^m\right)}^n=a^{m·n}.\hfill & & & \hfill \frac{\left(x^{12}\right)\left(x^{10}\right)}{\left(x^{30}\right)}\hfill \\ \text{Add the exponents in the numerator.}\hfill & & & \hfill \frac{x^{22}}{x^{30}}\hfill \\ \text{Use the Quotient Property,}\frac{a^m}{a^n}=\frac{1}{a^{n-m}}.\hfill & & & \hfill \frac{1}{x^8}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \hfill \frac{{\left(10p^3\right)}^2}{{\left(5p\right)}^3{\left(2p^5\right)}^4}\hfill \\ \text{Use the Product to a Power Property,}{\left(ab\right)}^m=a^m{b}^m.\hfill & & & \hfill \frac{{\left(10\right)}^2{\left(p^3\right)}^2}{{\left(5\right)}^3{\left(p\right)}^3{\left(2\right)}^4{\left(p^5\right)}^4}\hfill \\ \text{Use the Power Property,}{\left(a^m\right)}^n=a^{m·n}.\hfill & & & \hfill \frac{100p^6}{125p^3·16p^{20}}\hfill \\ \text{Add the exponents in the denominator.}\hfill & & & \hfill \frac{100p^6}{125·16p^{23}}\hfill \\ \text{Use the Quotient Property,}\frac{a^m}{a^n}=\frac{1}{a^{n-m}}.\hfill & & & \hfill \frac{100}{125·16p^{17}}\hfill \\ \text{Simplify.}\hfill & & & \hfill \frac{1}{20p^{17}}\hfill \end{array}$