# Questions on product of power rule of exponents

In this chapter we will discuss questions related to product of power rule with detailed solution.

The law of product of power rule is given as;

\mathtt{a^{m} \times a^{n} =\ a^{m+n}}

Practice the formula in paper so that you can remember it for long time.

Question 01
Solve the below expression

\mathtt{( i) \ 5\ \left( a^{2} b^{3}\right)^{4} \ \times 6\left( a^{4} b^{5}\right)^{2} \ }\\\ \\ \mathtt{( ii) \ \left( 11\ \times 10^{3}\right)\left( 4\times 10^{-7}\right)}\\\ \\ \mathtt{( iii) \ \left( ab^{5} c^{2}\right) \times 13\left( a^{-3} b^{-2} c^{11}\right)}\\\ \\ \mathtt{( iv) \ 7^{3} \times 9^{3}}\\\ \\ \mathtt{( v) \ 18^{9} \times 18^{-4}}

Solution

\mathtt{( i) \ 5\ \left( a^{2} b^{3}\right)^{4} \ \times 6\left( a^{4} b^{5}\right)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ 5\left( a^{4\times 2} .b^{3\times 4}\right) .\ 6\left( a^{4\times 2} .b^{5\times 2}\right)}\\\ \\ \mathtt{\Longrightarrow \ 5\left( a^{8} .b^{12}\right) .6\left( a^{8} .b^{10}\right)}\\\ \\ \mathtt{\Longrightarrow \ 30\ \left( a^{8+8}\right)\left( b^{12+10}\right)}\\\ \\ \mathtt{\Longrightarrow \ 30\ .a^{16} .\ b^{22}}

\mathtt{( ii) \ \left( 11\ \times 10^{3}\right)\left( 4\times 10^{-7}\right)}\\\ \\ \mathtt{\Longrightarrow \ ( 11\times 4)\left( 10^{3} \times 10^{-7}\right)}\\\ \\ \mathtt{\Longrightarrow \ 44\ .\ 10^{3-7}}\\\ \\ \mathtt{\Longrightarrow \ 44.\ 10^{-4}}

\mathtt{( iii) \ \left( ab^{5} c^{2}\right) \times 13\left( a^{-3} b^{-2} c^{11}\right)}\\\ \\ \mathtt{\Longrightarrow \ 13\ \left( a.a^{-3}\right)\left( b^{5} .b^{-2}\right)\left( c^{2} .c^{11}\right)}\\\ \\ \mathtt{\Longrightarrow \ 13\ \left( a^{1-3}\right)\left( b^{5-2}\right)\left( c^{2+11}\right)}\\\ \\ \mathtt{\Longrightarrow \ 13\ \left( a^{-2}\right)\left( b^{3}\right)\left( c^{13}\right)}\\\ \\ \mathtt{\Longrightarrow \ 13\ a^{-2} b^{3} c^{13}}

\mathtt{( iv) \ 7^{3} \times 9^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 7\times 9)^{3}}\\\ \\ \mathtt{\Longrightarrow \ 63^{3}}

\mathtt{( v) \ 18^{9} \times 18^{-4}}\\\ \\ \mathtt{\Longrightarrow \ 18^{9-4}}\\\ \\ \mathtt{\Longrightarrow \ 18^{5}}

Question 02
Solve the below expression;

\mathtt{( i) \ \left( x^{\frac{2}{3}} y^{\frac{-1}{2}}\right)^{4} \times \left( x^{6} y^{\frac{5}{6}}\right)}\\\ \\ \mathtt{( ii) \ \left(\sqrt{x^{3}}\right)^{4} \times \left( x^{-\frac{1}{2}}\right)}\\\ \\ \mathtt{( iii) \ 25^{\frac{5}{2}} \times 243^{\frac{3}{5}}}\\\ \\ \mathtt{( iv) \ \left(\frac{\sqrt{3}}{7}\right)^{4} \times \left(\frac{\sqrt{3}}{7}\right)^{2}}\\\ \\ \mathtt{( v) \ \sqrt{5\times 2^{-3}} \times 5^{4}}

Solution
\mathtt{( i) \ \left( x^{\frac{2}{3}} y^{\frac{-1}{2}}\right)^{4} \times \left( x^{6} y^{\frac{5}{6}}\right)}\\\ \\ \mathtt{\Longrightarrow \ \left( x^{\frac{2\times 4}{3}} y^{\frac{-4}{2}}\right) \times \left( x^{6} y^{\frac{5}{6}}\right)}\\\ \\ \mathtt{\Longrightarrow \ \left( x^{\frac{8}{3}} .y^{-2}\right) \times \left( x^{6} y^{\frac{5}{6}}\right)}\\\ \\ \mathtt{\Longrightarrow x^{\frac{8}{3} +6} .\ y^{-2+\frac{5}{6}}}\\\ \\ \mathtt{\Longrightarrow \ x^{\frac{26}{3}} .\ y^{\frac{-7}{6}}}

\mathtt{( ii) \ \left(\sqrt{x^{3}}\right)^{4} \times \left( x^{-\frac{1}{2}}\right)}\\\ \\ \mathtt{\Longrightarrow \ \left( x^{\frac{3}{2}}\right)^{4} \times x^{\frac{-1}{2}}}\\\ \\ \mathtt{\Longrightarrow \ x^{\frac{3}{2} \times 4} \times x^{\frac{-1}{2}}}\\\ \\ \mathtt{\Longrightarrow \ x^{6} \times x^{\frac{-1}{2}}}\\\ \\ \mathtt{\Longrightarrow \ x^{6-\frac{1}{2}}}\\\ \\ \mathtt{\Longrightarrow \ x^{\frac{11}{2}}}

\mathtt{( iii) \ 25^{\frac{5}{2}} \times 243^{\frac{3}{5}}}\\\ \\ \mathtt{\Longrightarrow \ \left( 5^{2}\right)^{\frac{5}{2}} \times \left( 3^{5}\right)^{\frac{3}{5}}}\\\ \\ \mathtt{\Longrightarrow \ 5^{2\times \frac{5}{2}} \times 3^{5\times \frac{3}{5}}}\\\ \\ \mathtt{\Longrightarrow \ 5^{5} \times 3^{3}}\\\ \\ \mathtt{\Longrightarrow \ 84375}

\mathtt{( iv) \ \left(\frac{\sqrt{3}}{7}\right)^{4} \times \left(\frac{\sqrt{3}}{7}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \ \left(\frac{\sqrt{3}}{7}\right)^{4+2}}\\\ \\ \mathtt{\Longrightarrow \ \ \left(\frac{\sqrt{3}}{7}\right)^{6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\left( 3^{\frac{1}{2}}\right)^{6}}{7^{6}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3^{3}}{7^{6}}}

\mathtt{( v) \ \sqrt{5\times 2^{-3}} \times 5{^{4}}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt{\frac{5}{2^{3}}} \times 5^{4}}\\\ \\ \mathtt{\Longrightarrow \ 5^{\frac{1}{2}} \times \frac{1}{\sqrt{2^{3}}} \times 5^{4}}\\\ \\ \mathtt{\Longrightarrow \ 5^{\frac{1}{2} +4} \times \frac{1}{\sqrt{2^{3}}}}\\\ \\ \mathtt{\Longrightarrow 5^{\frac{9}{2}} \times 8^{\frac{-1}{2}}}