Index Laws in Algebra for ESAT Mathematics

Index Notation

In algebra, we use index notation to represent repeated multiplication of the same factor. For the expression $a^5$, we say that $a$ is raised to the power of 5. In this context, $a$ is known as the base and 5 is the power or index (the plural form is indices).

The expression $a^5$ is shorthand for:
$a \times a \times a \times a \times a = a^5$

Multiplication

When multiplying powers that have the same base, you must add the indices together. The general rule is:
$a^m \times a^n = a^{m+n}$

If the algebraic terms have numerical coefficients, these coefficients are multiplied together separately from the variables. For example, in the term $5p$, 5 is the coefficient. Consider the following simplification:
$5p^2q \times 2pq^2 = 10p^3q^3$

You should be able to multiply various powers of the same base to simplify complex expressions effectively.

Division

To divide powers of the same base, you subtract the index of the divisor from the index of the dividend. The general rule is:
$a^m \div a^n = a^{m-n}$

As with multiplication, any numerical coefficients must be divided first. For example:
$20p^3q^2 \div 2p^2q = \frac{20}{2}p^{3-2}q^{2-1} = 10pq$

You are expected to be proficient in dividing powers of the same base to reduce expressions to their simplest form.

Raising Terms to a Further Power

When a term that already contains a power is raised to another power, you multiply the indices together. The general rule is:
$(a^m)^n = a^{m \times n} = (a^n)^m$

It is crucial to apply the power to every component within the bracket, including numerical coefficients and all variables. For example:
$(4ab^2)^3 = 4^3a^3b^{2 \times 3} = 64a^3b^6$

Zero and Unitary Powers

There are two specific cases that appear frequently:

1. Any non-zero base raised to the power of zero is equal to 1. In general, $a^0 = 1$ for all non-zero values of $a$.
2. Any base raised to the power of 1 is simply the base itself, so $a^1 = a$.

Powers of Fractions

When a fraction is raised to a power, the power applies to both the numerator and the denominator. The general rule is:
$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$

You must be able to raise fractions to a power and simplify the resulting expression.

Negative Powers

A base raised to a negative power is equivalent to the reciprocal of the base raised to the corresponding positive power. The general rule is:
$a^{-m} = \frac{1}{a^m}$

You should be able to apply this to bases where the coefficient is not 1, ensuring the coefficient stays in the correct position.

Fractional Powers

Fractional indices represent roots of a number or variable.

1. The power $\frac{1}{2}$ is the square root: $a^{\frac{1}{2}} = \sqrt{a}$
2. The power $\frac{1}{3}$ is the cube root: $a^{\frac{1}{3}} = \sqrt[3]{a}$
3. The power $\frac{1}{4}$ is the fourth root: $a^{\frac{1}{4}} = \sqrt[4]{a}$

In general, $a^{\frac{1}{b}} = \sqrt[b]{a}$. Furthermore, a negative fractional power is written as:
$a^{-\frac{1}{b}} = \frac{1}{\sqrt[b]{a}}$

For more complex fractions, the numerator represents a power and the denominator represents a root:
$a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m} = (a^{\frac{1}{n}})^m = (\sqrt[n]{a})^m$

Worked Examples

Example 1: Multiplying powers
Simplify $4a^2 \times 2a^3$
Multiply the coefficients and add the indices:
$4 \times 2 \times a^{3+2} = 8a^5$

Example 2: Dividing powers
Simplify $12a^6 \div 2a^3$
Divide the numerical coefficients and subtract the indices:
$6a^{6-3} = 6a^3$

Example 3: Raising a power to a power
Simplify $(2p^3)^5$
Apply the power to the coefficient and multiply the variable's index:
$2^5p^{3 \times 5} = 32p^{15}$

Example 4: Raising fractions to a power
Simplify $\left(\frac{2p^2}{3q^3}\right)^4$
Apply the power to every term in the numerator and denominator:
$\frac{(2p^2)^4}{(3q^3)^4} = \frac{2^4 p^{2 \times 4}}{3^4 q^{3 \times 4}} = \frac{16 p^8}{81 q^{12}}$

Example 5: Negative powers
Simplify $(4q^3)^{-4}$
Write as a reciprocal with a positive power:
$\frac{1}{4^4 q^{3 \times 4}} = \frac{1}{256q^{12}}$ or $\frac{1}{256} q^{-12}$

Example 6: Fractional powers
Simplify $(25y^4)^{\frac{1}{2}}$
Take the square root of the coefficient and multiply the index by $\frac{1}{2}$:
$25^{\frac{1}{2}} (y^4)^{\frac{1}{2}} = 5y^{4 \times \frac{1}{2}} = 5y^2$

Example 7: Negative and fractional powers
Simplify $(8a^{-3})^{\frac{1}{3}}$
Take the cube root of the coefficient and multiply the index by $\frac{1}{3}$:
$8^{\frac{1}{3}} (a^{-3})^{\frac{1}{3}} = 2a^{-1} = \frac{2}{a}$