Number system practice

Rules of exponents & indices

Question: Simplify and find the value of

(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)(6/6) = 3

Rule of exponents used: = axy

(b) \{(64)}^{\frac{2}{3}})

Prime factorize 64 = 2 * 2 * 2 * 2 * 2 * 2 = 26
\{(64)}^{\frac{2}{3}}) = = \{(2)}^{\frac{12}{3}}) = 24 = 16.

Rule of exponents used: = axy

(c) \{(243)}^{\frac{6}{5}})

Prime factorize 243 = 3 * 3 * 3 * 3 * 3 = 35
\{(243)}^{\frac{6}{5}}) = = 36 = 729

Rule of exponents used: = 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