Index Laws for Numerical Expressions

Index Numbers and Powers

The power to which a number is raised is known as the index, or the power. When we write $2 \times 2 \times 2 \times 2 \times 2 = 2^5 = 32$, we say that $2^5$ is 32 expressed in index form. Students preparing for university admission must be able to convert numbers into index form and evaluate them accurately.

Multiplication and Division Laws

For powers that share the same base, there are three primary laws to master.

Multiplication

To multiply powers with the same base, you add the indices together. The general rule is: $a^m \times a^n = a^{m+n}$

It is also important to note the related rule for products raised to a power: $(ab)^n = a^n b^n$

Division

To divide powers with the same base, you subtract the indices. The general rule is: $a^m \div a^n = a^{m-n}$

Special Indices: Zero and Unity

There are specific results for the indices 0 and 1 that must be memorised:

1. Any non:zero number raised to the power of 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 simply the number itself. In general, $a^1 = a$.
3. The number 1 raised to any power remains 1.

Fractions and Negative Powers

Fractions raised to a power

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

Negative powers

A number raised to a negative power represents the reciprocal of that number raised to the corresponding positive power: $a^{-m} = \frac{1}{a^m}$

Raising Powers and Fractional Indices

Power of a power

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

Fractional powers

Fractional indices represent roots. Specifically:

1. The power $\frac{1}{2}$ is equivalent to the square root.
2. The power $\frac{1}{3}$ is equivalent to the cube root.
3. The power $\frac{1}{4}$ is equivalent to the fourth root.

The general form for fractional powers is: $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Worked Examples

Basic Conversions and Evaluations

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

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

Using Multiplication and Division Laws

Write as a single power of 5: $5^3 \times 5^4$ By adding the powers: $5^{3+4} = 5^7$

Write as a single power of 2: $2^8 \div 2^5$ By subtracting the powers: $2^{8-5} = 2^3$

Combining Multiple Operations

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

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

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

Advanced Index Law Problems

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

Evaluate $49^{-\frac{1}{2}}$: First handle the negative: $\frac{1}{49^{\frac{1}{2}}}$. Then the fractional index: $\frac{1}{\sqrt{49}} = \frac{1}{7}$

Evaluate $16^{-\frac{3}{4}}$: Rewrite as $\frac{1}{16^{\frac{3}{4}}}$. Apply the root first: $\frac{1}{(\sqrt[4]{16})^3}$. Since $\sqrt[4]{16} = 2$, we have $\frac{1}{2^3} = \frac{1}{8}$.