Form 3 Mathematics Bab 1: Indices
Repeated multiplication of a number can be written in index form as $a^n$, where:
$$\text{Repeated Multiplication: } \underbrace{a \times a \times a \times \dots \times a}_{n \text{ times}} = a^n$$
When multiplying terms with the same base, add the indices:
$$a^m \times a^n = a^{m + n}$$Example: $3^4 \times 3^5 = 3^{4+5} = 3^9$
When dividing terms with the same base, subtract the indices:
$$a^m \div a^n = a^{m - n}$$Example: $m^8 \div m^3 = m^{8-3} = m^5$
When an index term is raised to a power, multiply the indices:
$$(a^m)^n = a^{m \times n} = a^{mn}$$Example: $(5^3)^4 = 5^{3 \times 4} = 5^{12}$
For product or quotient of terms raised to a power:
$$(a^m \times b^n)^p = a^{mp} \times b^{np}$$ $$\left(\frac{a^m}{b^n}\right)^p = \frac{a^{mp}}{b^{np}}$$Any non-zero number raised to the power of zero is equal to 1:
$$a^0 = 1 \quad (a \neq 0)$$Example: $8^0 = 1, \quad (-12)^0 = 1, \quad (5x)^0 = 1$
A term with a negative index is equal to the reciprocal of the term with a positive index:
$$a^{-n} = \frac{1}{a^n} \quad \text{and} \quad \frac{1}{a^{-n}} = a^n \quad (a \neq 0)$$Example: $4^{-3} = \frac{1}{4^3} = \frac{1}{64}, \quad \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$
A term with a fractional index represents roots and powers:
$$a^{\frac{1}{n}} = \sqrt[n]{a}$$ $$a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m} \quad \text{or} \quad a^{\frac{m}{n}} = (a^{\frac{1}{n}})^m = (\sqrt[n]{a})^m$$Example: $27^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9$
| Law | Formula | Example |
|---|---|---|
| Multiplication | $a^m \times a^n = a^{m+n}$ | $x^2 \times x^3 = x^5$ |
| Division | $a^m \div a^n = a^{m-n}$ | $y^7 \div y^2 = y^5$ |
| Power | $(a^m)^n = a^{mn}$ | $(2^3)^2 = 2^6 = 64$ |
| Zero Index | $a^0 = 1$ | $100^0 = 1$ |
| Negative Index | $a^{-n} = \frac{1}{a^n}$ | $3^{-2} = \frac{1}{9}$ |
| Fractional Index | $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ | $16^{\frac{3}{4}} = (\sqrt[4]{16})^3 = 2^3 = 8$ |