Indicial Equations & Standard Form of Numbers - SS1 Mathematics Lesson Note

Indicial Equations

Example 3 Solve the following equations: (a) \(2^{x} = 64\) (b) \(3^{3x} = \ 81^{\frac{3}{4}}\)

Solution (a) \(2^{x} = 64\ \rightarrow \ 2^{x} = 2^{6}\ \rightarrow Compare\ the\ powers,\ thus,\ x = 6\ \)

(b) \(3^{3x} = \ 81^{\frac{3}{4}}\ \rightarrow \ 3^{3x} = \ \sqrt[4]{81^{3}}\ \rightarrow \ 3^{3x} = \ \sqrt[4]{{(3^{4})}^{3}}\ \rightarrow \ 3^{3x} = \ \sqrt[4]{3^{4 \times 3}}\ \ \rightarrow \ 3^{3x} = \ 3^{\frac{4 \times 3}{4}}\ \rightarrow \ 3^{3x} = \ 3^{3}\ \rightarrow comparing\ the\ powers,\ 3x = 3\ \rightarrow x = \frac{3}{3} = 1\ \)

Standard Form of Numbers

A number in standard form is of the form \(a \times 10^{n}\), where \(\mathbf{a}\) can be a whole positive or negative number (between \(0\ and\ 10\) or \(0\ and\ - 10\)), or a decimal positive or negative number; \(\mathbf{n}\) can be a whole positive or negative number or zero.

Example 4 Write in their standard forms the numbers \(156,000\) and \(0.0000032\)

Solution (a) \(156,000 = 1.56\ \times 100000 = 1.56\ \times \ 10^{5}\) moving the decimal point to between the first two non-zero digits

For numbers > 1, the index of the power 10 is positive

(b) \(0.0000032 = \frac{3.2}{1000000} = 3.2{\times 10}^{- 6}\) moving the decimal point to between the first two non-zero digits

For numbers < 1, the index of the power 10 is negative