# Problems on quotient of power exponents

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}