In this chapter we will discuss problems related to quotient of power rule with complete solution.
The law for quotient for power rule is given as;
\mathtt{a^{m} \div a^{n} =\ a^{m-n}}
Question
Simplify the below expressions;
(i) \mathtt{\frac{x^{2} y^{5}}{x^{6} y^{3}}}
(ii) \mathtt{\frac{\left( 4x^{3}\right)^{2}}{x^{4}}}
(iii) \mathtt{\left(\frac{x^{-3}}{x^{-2}}\right)^{\frac{5}{4}}}
(iv) \mathtt{\left(\frac{\sqrt{3}}{8}\right)^{5} \div \ \left(\frac{\sqrt{3}}{8}\right)^{9}}
(v) \mathtt{\frac{( 25)^{\frac{3}{2}}}{( 125)^{\frac{2}{3}}}}
Solution
(i) \mathtt{\frac{x^{2} y^{5}}{x^{6} y^{3}}}
\mathtt{\Longrightarrow \ x^{2-6} \ y^{5-3}}\\\ \\ \mathtt{\Longrightarrow \ x^{-4} .y^{2}}
(ii) \mathtt{\frac{\left( 4x^{3}\right)^{2}}{x^{4}}}
\mathtt{\Longrightarrow \ \frac{16\ x^{3\times 2}}{x^{4}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{16x^{6}}{x^{4}}}\\\ \\ \mathtt{\Longrightarrow \ 16x^{6-4}}\\\ \\ \mathtt{\Longrightarrow \ 16x^{2}}
(iii) \mathtt{\left(\frac{x^{-3}}{x^{-2}}\right)^{\frac{5}{4}}}
\mathtt{\Longrightarrow \ \left( x^{-3-( -2)}\right)^{\frac{5}{4}}}\\\ \\ \mathtt{\Longrightarrow \ \left( x^{-1}\right)^{\frac{5}{4}}}\\\ \\ \mathtt{\Longrightarrow \ x^{\frac{-5}{4}}}
(iv) \mathtt{\left(\frac{\sqrt{3}}{8}\right)^{5} \div \ \left(\frac{\sqrt{3}}{8}\right)^{9}}
\mathtt{\Longrightarrow \left(\frac{\sqrt{3}}{8}\right)^{5-9}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{\sqrt{3}}{8}\right)^{-4}}
(v) \mathtt{\frac{( 25)^{\frac{3}{2}}}{( 125)^{\frac{2}{3}}}}
\mathtt{\Longrightarrow \ \frac{\left( 5^{2}\right)^{\frac{3}{2}}}{\left( 5^{3}\right)^{\frac{2}{3}}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5^{3}}{5^{2}}}\\\ \\ \mathtt{\Longrightarrow \ 5^{3-2}}\\\ \\ \mathtt{\Longrightarrow \ 5}