# Number system practice

Rules of exponents & indices

(a) ${$729)}^{$frac{1}{6}})$b) ${$64)}^{$frac{2}{3}})$c) ${$243)}^{$frac{6}{5}})$d) ${$21)}^{$frac{3}{2}} $times {$21$}^{$frac{5}{2}})$e) $$frac{{$81$}^{$frac{1}{3}}}{{$81)}^{$frac{1}{12}}}) #### Video Explanation for all 5 questions #### Explanatory Answer ####$a) ${$729)}^{$frac{1}{6}}) Prime factorize 729 = 3 * 3 * 3 * 3 * 3 * 3 = 36 ${$729$}^{$frac{1}{6}}) = ${$3}^{{6)}^{1/6}}$ = (3)(6/6) = 3

Rule of exponents used: ${(a}^{{x)}^{y}}$ = axy

#### (c) ${$243)}^{$frac{6}{5}}) Prime factorize 243 = 3 * 3 * 3 * 3 * 3 = 35 ${$243$}^{$frac{6}{5}}) = ${$3}^{{5)}^{6/5}}$ = 36 = 729

Rule of exponents used: ${(a}^{{x)}^{y}}$ = axy

#### (d) ${$21)}^{$frac{3}{2}} $times {$21$}^{$frac{5}{2}}) ${$21$}^{$frac{3}{2}} $times {$21$}^{$frac{5}{2}}) = ${$21$}^{$frac{3}{2} + $frac{5}{2}}$ = ${$21$}^{$frac{8}{2}}) = 214 Rule of exponents used: ax * ay = ax + y ####$e) $$frac{{$81$}^{$frac{1}{3}}}{{$81)}^{$frac{1}{12}}}) $\frac{{$81$}^{$frac{1}{3}}}{{$81)}^{$frac{1}{12}}}) = ${$81$}^{$frac{1}{3} - $frac{1}{12}}$ = ${$81$}^{$frac{{4-1}}{12}}) = ${$81$}^{$frac{3}{12}}) = ${$81$}^{$frac{1}{4}}) Prime factorize 81 = 3 * 3 * 3 * 3 = 34 ${$81$}^{$frac{1}{4}}) = ${${3^4}$}^{$frac{1}{4}}) = ${$3$}^{$frac{4}{4}}$ = 3

Rule of exponents used: $\frac{a^x}{a^y}$ = ax – y