4.1.2 Indices and Laws of Indices (Part 2) - SPM Additional Mathematics

4.1 Indices and Laws of Indices (Part 2)

(C) Fractional Indices

\begin{array}{l} a^{\frac{1}{n}} is a n th root of a . \\ a^{\frac{1}{n}} = \sqrt[n]{a} \\ a^{\frac{m}{n}} is a n th root of a^{m} . \\ a^{\frac{m}{n}} = \sqrt[n]{a^{m}} \end{array}

Example 1:

Find the value of the followings:

\begin{array}{l} {(a) 81}^{\frac{1}{2}} \\ {(b) 64}^{\frac{1}{3}} \\ {(c) 625}^{\frac{1}{4}} \end{array}

Solution:

\begin{array}{l} {(a) 81}^{\frac{1}{2}} = \sqrt{81} = 9 \\ {(b) 64}^{\frac{1}{3}} = \sqrt[3]{64} = 4 \\ {(c) 625}^{\frac{1}{4}} = \sqrt[4]{625} = 5 \end{array}

Example 2:

Find the value of the followings:

\begin{array}{l} (a) 16^{\frac{3}{2}} \\ (b) {(\frac{27}{64})}^{\frac{2}{3}} \end{array}

Solution:

\begin{array}{l} (a) 16^{\frac{3}{2}} = {(16^{\frac{1}{2}})}^{3} = 4^{3} = 64 \\ (b) {(\frac{27}{64})}^{\frac{2}{3}} = {(\sqrt[3]{\frac{27}{64}})}^{2} = {(\frac{3}{4})}^{2} = \frac{9}{16} \end{array}

(D) Laws of Indices

\begin{array}{l} a^{m} \times a^{n} = a^{m + n} \\ Example: \\ 3^{3} \times 3^{2} \\ = 3^{3 + 2} = 3^{5} = 243 \end{array}

\begin{array}{l} a^{m} \div a^{n} = a^{m - n} or \frac{a^{m}}{a^{n}} = a^{m - n}, a \neq 0 \\ Example: \\ 3^{3} \div 3^{2} \\ = 3^{3 - 2} = 3^{1} = 3 \\ or \\ \frac{3^{3}}{3^{2}} = 3^{3 - 2} = 3^{1} = 3 \end{array}

\begin{array}{l} {(a^{m})}^{n} = a^{m n} \\ Example: \\ {(7^{3})}^{4} = 7^{3 \times 4} = 7^{12} \end{array}

\begin{array}{l} {(a b)}^{n} = a^{n} b^{n} \\ Example: \\ {(15)}^{3} = {(5 \times 3)}^{3} = 2^{3} \times 3^{3} \end{array}

\begin{array}{l} {(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}, b \neq 0 \\ Example: \\ {(\frac{3}{5})}^{4} = \frac{3^{4}}{5^{4}} = \frac{81}{625} \end{array}