Index Laws and Powers for ESAT Mathematics 1

Index Numbers or Powers

The power to which a number is raised is called the index (plural: indices). For example, $2 \times 2 \times 2 \times 2 \times 2 = 2^5 = 32$. Here, $2^5$ is 32 written in index form. You should be able to convert numbers into index form and evaluate numbers already written in index form.

Example: Writing a number in index form

Write 243 as a power of 3. $243 = 3 \times 3 \times 3 \times 3 \times 3 = 3^5$

Example: Evaluating index form

Evaluate $2^7$. $2^7 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128$

The Index Laws

For powers involving the same base number, the following laws allow for easy simplification of expressions.

Multiplication

To multiply powers of the same number, add the indices together: $a^m \times a^n = a^{m+n}$ Note also that when a product is raised to a power, the power applies to each factor: $(ab)^n = a^n b^n$.

Example: Multiplying powers

Write as a single power of 5: $5^3 \times 5^4$ Remembering to add the powers gives: $5^{3+4} = 5^7$

Division

To divide powers of the same number, subtract the indices: $a^m \div a^n = a^{m-n}$

Example: Dividing powers

Write as a single power of 2: $2^8 \div 2^5$ Remembering to subtract the powers gives: $2^{8-5} = 2^3$

Powers of Zero and One

There are specific rules for evaluating indices of zero and one:

1. Any number raised to the power 0 is equal to 1. In general, $a^0 = 1$ for all non-zero values of $a$.
2. Any number raised to the power of 1 is just the number itself. In general, $a^1 = a$.
3. The number 1 raised to any power is always 1.

Example: Combining base rules

Evaluate $(1.2)^0 + 5^3 + 1^5 - 7^1$ $(1.2)^0 + 5^3 + 1^5 - 7^1 = 1 + 125 + 1 - 7 = 120$

Fractions

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

Example: Evaluating fractional bases

Work out $\left(\frac{3}{4}\right)^4$ Raise both the numerator and the denominator of the fraction to the power outside the brackets: $\left(\frac{3}{4}\right)^4 = \frac{3^4}{4^4} = \frac{81}{256}$

Negative Powers

A number raised to a negative power can be written as 1 over the number raised to the corresponding positive power: $a^{-m} = \frac{1}{a^m}$

Example: Evaluating negative powers

Evaluate $5^{-3}$ $5^{-3} = \frac{1}{5^3} = \frac{1}{125}$

Raising Powers to a Further Power

To raise a power to a further power, multiply the powers together: $(a^m)^n = a^{mn}$

Example: Nested powers

Work out $(2^3)^{-2}$ Remember to multiply the powers: $(2^3)^{-2} = 2^{3 \times -2} = 2^{-6} = \frac{1}{2^6} = \frac{1}{64}$

Fractional Powers

Fractional powers represent roots of the base number:

1. The power $\frac{1}{2}$ is the same as the square root: $a^{1/2} = \sqrt{a}$
2. The power $\frac{1}{3}$ is the same as the cube root: $a^{1/3} = \sqrt[3]{a}$
3. The power $\frac{1}{4}$ is the same as the fourth root, and so on.

In general, for a fractional power with a numerator greater than 1: $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Example: Negative fractional power

Evaluate $49^{-\frac{1}{2}}$ $49^{-\frac{1}{2}} = \frac{1}{49^{\frac{1}{2}}} = \frac{1}{\sqrt{49}} = \frac{1}{7}$

Example: Complex fractional power

Evaluate $16^{-\frac{3}{4}}$ $16^{-\frac{3}{4}} = \frac{1}{16^{\frac{3}{4}}} = \frac{1}{(\sqrt[4]{16})^3} = \frac{1}{2^3} = \frac{1}{8}$