State and apply the laws of indices, including zero, negative and fractional indices, to simplify expressions and solve simple index equations

What this dot point is asking

SEAB wants you to know the laws of indices and use them fluently to simplify expressions involving powers, including zero, negative and fractional indices. You also need to solve simple index equations such as $2^x = 32$ by writing both sides with the same base. Indices are the language of growth and decay and the foundation for logarithms, so this is one of the most reused skills in the whole syllabus.

The answer

The core laws

For the same base $a$ (with $a \ne 0$):

When you multiply like bases you add indices; when you divide you subtract; a power of a power multiplies. Powers also distribute over a product: $(ab)^n = a^n b^n$.

Zero and negative indices

A zero index gives $1$; a negative index means "one over". For example $5^0 = 1$ and $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$.

Fractional indices

A fractional index is a root combined with a power:

So $9^{1/2} = \sqrt{9} = 3$ and $8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$. Taking the root first usually keeps the numbers small.

Rewriting roots and reciprocals as indices

A great deal of algebra becomes easier once roots and reciprocals are written as indices, because the three core laws can then be applied directly. The standard conversions are:

For example, $\dfrac{x^2}{\sqrt{x}} = \dfrac{x^2}{x^{1/2}} = x^{2 - 1/2} = x^{3/2}$. Converting first turns an awkward-looking fraction into a single power that obeys the subtraction law cleanly.

Solving simple index equations

If you can write both sides of an equation as powers of the same base, then the indices must be equal:

This works because the exponential function is one-to-one. When the bases cannot be matched, logarithms are used instead (covered in the exponential and logarithmic strand).

Examples in context

Example 1. Compound growth. A sum of money growing at a fixed rate is modelled by $A = P(1.05)^n$ after $n$ years. Manipulating that power, and later solving for $n$, depends entirely on the index laws, which is why indices lead directly into the logarithm topic.

Example 2. Simplifying before differentiating. Calculus questions often hide a power, for example $\dfrac{1}{\sqrt{x}} = x^{-1/2}$. Rewriting roots and reciprocals as indices first is what makes the power rule for differentiation applicable, so this skill resurfaces in the calculus strand.

Try this

Q1. Simplify $(x^3)^2 \times x$. [2 marks]

• Cue. $(x^3)^2 = x^6$, then $x^6 \times x = x^7$.

Q2. Evaluate $27^{2/3}$. [2 marks]

• Cue. $\left(\sqrt[3]{27}\right)^2 = 3^2 = 9$.

Q3. Solve $3^{x} = \dfrac{1}{9}$. [2 marks]

• Cue. $\dfrac{1}{9} = 3^{-2}$, so $x = -2$.