Laws of Indices - SS1 Mathematics Lesson Note

1. Multiplication (Product): \(N^{a} \times N^{b} = \ N^{a + b}\)

Example \(3^{2} \times 3^{8} = \ 3^{2 + 8} = \ 3^{10}\)

1. Division (Quotient): \(N^{a} \div N^{b} = \ N^{a - b}\)

Example \(3^{2} \div 3^{8} = \ 3^{2 - 8} = \ 3^{- 6}\)

1. Powers: \({(N^{a})}^{\mathbf{b}} = \ N^{a \times b} = \ N^{ab}\)

Example \({(x^{4})}^{6} = \ x^{4 \times 6} = \ x^{24}\)

1. Fractional Indices (powers and roots): \(N^{\frac{a}{b}} = \ \sqrt[b]{N^{a}}\)

Example \(8^{\frac{2}{3}} = \ \sqrt[3]{8^{2}} = \ \sqrt[3]{64} = 4\)

1. Zero Index: \(N^{0} = 1\) (because \(N^{0} = N^{a - a} = N^{a} \div N^{a} = \frac{N^{a}}{N^{a}} = 1\))

Example \(y^{0} = 1\) and \(8^{0} = 1\)

1. Negative Indices \(N^{- a} = \ \frac{1}{N^{a}}\)

Example \(3^{- 6} = \ \frac{1}{3^{6}} = \ \frac{1}{729}\)