1.
Simplify by using the rules of exponents and name the rules used.
(a)
$\displaystyle \frac{{36{{a}^{4}}{{b}^{5}}}}{{100{{a}^{7}}{{b}^{3}}}}$
Show/Hide Solution
$\begin{aligned}
&\ \ \ \ \ \displaystyle \frac{{36{{a}^{4}}{{b}^{5}}}}{{100{{a}^{7}}{{b}^{3}}}}\\
&=\displaystyle\frac{9}{{25}}\times \frac{1}{{{{a}^{{7-4}}}}}\times {{b}^{{5-3}}}\ \ \ \ \ (\text{Division Rule})\\
&=\displaystyle \frac{{9{{b}^{2}}}}{{25{{a}^{3}}}}
\end{aligned}$
(b)
$\displaystyle \frac{27 a^{2} b^{5}}{\left(9 a^{2} b\right)^{2}}$
Show/Hide Solution
$\begin{aligned}
&\ \ \ \ \ \displaystyle \frac{27 a^{2} b^{5}}{\left(9 a^{2} b\right)^{2}}\\
&=\displaystyle \frac{{27{{a}^{2}}{{b}^{5}}}}{{81{{a}^{4}}{{b}^{2}}}}\ \ \ \ \ (\text{Power of a Powar Rule})\\
&=\displaystyle \frac{1}{3}\times \frac{1}{{{{a}^{{4-2}}}}}\times {{b}^{{5-2}}}\ \ \ \ \ (\text{Division Rule})\\
&=\displaystyle \frac{{{{b}^{3}}}}{{3{{a}^{2}}}}
\end{aligned}$
(c)
$\displaystyle \left(\frac{-135 a^{4} b^{5} c^{6}}{315 a^{6} b^{7} c^{8}}\right)^{2}$
Show/Hide Solution
$\begin{aligned}
&\ \ \ \ \ \displaystyle {{\left( {\frac{{-135{{a}^{4}}{{b}^{5}}{{c}^{6}}}}{{315{{a}^{6}}{{b}^{7}}{{c}^{8}}}}} \right)}^{2}}\\
&=\displaystyle {{\left( {\frac{{-3}}{{7{{a}^{{6-4}}}{{b}^{{7-2}}}{{c}^{{8-6}}}}}} \right)}^{2}}\ \ \ (\text{Division Rule})\\
&=\displaystyle \frac{{{{{(-3)}}^{2}}}}{{{{{(7)}}^{2}}{{{({{a}^{2}})}}^{2}}({{b}^{5}}){{{({{c}^{2}})}}^{2}}}}\ \ (\text{Power of a Quotient Rule})\\
&= \displaystyle \frac{9}{{49{{a}^{4}}{{b}^{5}}{{c}^{4}}}}\ \ (\text{Power of a Power Rule})
\end{aligned}$
(d)
$\displaystyle \left(\frac{x^{4}}{y^{5}}\right)^{3}\left(\frac{y^{3}}{x^{2}}\right)^{2}$
Show/Hide Solution
$\begin{aligned}
&\ \ \ \ \ \displaystyle {{\left( {\frac{{{{x}^{4}}}}{{{{y}^{5}}}}} \right)}^{3}}{{\left( {\frac{{{{y}^{3}}}}{{{{x}^{2}}}}} \right)}^{2}}\\
&=\displaystyle \left( {\frac{{{{x}^{{12}}}}}{{{{y}^{{15}}}}}} \right)\left( {\frac{{{{y}^{6}}}}{{{{x}^{4}}}}} \right)\ \ (\text{Power of a Quotient Rule})\\
&=\displaystyle \frac{{{{x}^{8}}}}{{{{y}^{9}}}}\ \ (\text{Division Rule})\\
\end{aligned}$
(e)
$\displaystyle \frac{2^{3^{2}}}{\left(2^{2}\right)^{3}}$
Show/Hide Solution
$\begin{aligned}
&\ \ \ \ \ \displaystyle \frac{{{{2}^{{{{3}^{2}}}}}}}{{{{{\left( {{{2}^{2}}} \right)}}^{3}}}}\\
&=\displaystyle \frac{{{{2}^{9}}}}{{{{2}^{6}}}}\ (\text{Power of a Power Rule})\\
&={{2}^{3}}\ \ (\text{Division Rule})\\
&=8
\end{aligned}$
2.
Evaluate the followings.
(a)
$\displaystyle \frac{54^{2} \times 12^{3} \times 64^{2}\left(3^{2} \times 4^{3} \times 5^{2}\right)^{3}}{\left(3^{2} \times 15 \times 20^{3}\right)^{4}}$
3.
Simplify.
(a)
$\displaystyle \left(\frac{3^{m}}{15^{n}}\right)^{3}\left(\frac{45^{n}}{255^{m}}\right)^{2}$
(b)
$\displaystyle \left(\frac{20^{x}}{400^{y}}\right)^{2}\left(\frac{150^{y^{2}}}{180^{x}}\right)^{3}$
(c)
$\displaystyle \frac{\left(x^{3}-y^{3}\right)(x+y)}{\left(x^{2}-y^{2}\right)^{3}}$
(d)
$\displaystyle \frac{\left(x^{a-b} x^{b-c}\right)^{a}\left(\frac{x^{a}}{x^{c}}\right)^{c}}{\left(x^{b} x^{c}\right)^{a} \div\left(x^{a+c}\right)^{c}}$
4.
Evaluate the followings.
5.
Simplify the followings.