Indices and the Laws of Exponents

Before you begin

You should be comfortable multiplying and dividing numbers and simplifying fractions.

The intuition

Writing $a^3$ means $a \times a \times a$ — three copies of $a$ multiplied together. The laws of indices are shortcuts that follow from this definition.

Definitions

• Base: the number being raised to a power.
• Index (exponent): the number of times the base is multiplied.
• $a^0 = 1$ (for $a \neq 0$).
• $a^{-n} = \frac{1}{a^n}$.
• $a^{1/n} = \sqrt[n]{a}$ (the $n$-th root).

The laws of indices

1. Multiplication: $a^m \times a^n = a^{m+n}$
2. Division: $a^m \div a^n = a^{m-n}$
3. Power of a power: $(a^m)^n = a^{mn}$
4. Power of a product: $(ab)^n = a^n b^n$
5. Power of a quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
6. Negative index: $a^{-n} = \frac{1}{a^n}$
7. Fractional index: $a^{m/n} = \left(\sqrt[n]{a}\right)^m$

Step-by-step method

1. Identify the base. The laws only apply when the base is the same.
2. Decide which law matches the operation.
3. Apply the law to combine or simplify the indices.
4. Give the answer with a single index where possible.

Worked examples

Foundation example

Simplify $x^5 \times x^3$. Same base, multiplication: add the indices. $x^5 \times x^3 = x^{5+3} = x^8$

Developing example

Simplify $\frac{6a^7 b^2}{2a^3 b^5}$ and write your answer with positive indices. Deal with numbers and each letter separately. $\frac{6a^7 b^2}{2a^3 b^5} = 3 \cdot a^{7-3} \cdot b^{2-5} = 3a^4 b^{-3} = \frac{3a^4}{b^3}$

Advanced example

Simplify $\left(2x^{-2}\right)^3 \times \left(x^5\right)^2$ with positive indices. $\left(2x^{-2}\right)^3 = 2^3 \cdot x^{-6} = 8x^{-6}$ $\left(x^5\right)^2 = x^{10}$ $8x^{-6} \times x^{10} = 8x^{-6+10} = 8x^4$

Common mistakes

• Different bases: $x^3 \times y^4$ cannot be simplified by adding indices.
• Forgetting brackets: $(2x)^3 = 8x^3$, not $2x^3$.
• Mixing up signs: $a^{-n}$ is a reciprocal, not a negative number.

Concept checklist

• I can apply each of the seven laws.
• I can convert between negative and fractional indices.
• I can simplify expressions with mixed indices.

Practice questions

1. Simplify $a^4 \times a^9$.
2. Simplify $\frac{x^{10}}{x^4}$.
3. Simplify $(y^3)^5$.
4. Evaluate $5^{-2}$.
5. Simplify $\frac{8x^6 y^3}{4x^2 y^7}$ with positive indices.
6. Simplify $(3a^{-2})^2 \times a^5$ with positive indices.

Challenge questions

1. Simplify $\frac{(2x^3)^4}{x^5 \cdot x^2}$ with positive indices.
2. Show that $16^{-3/4} = \frac{1}{8}$.

Frequently asked questions

What is a negative index? A negative index means the reciprocal of the positive power: $a^{-n} = \frac{1}{a^n}$.

Why is $a^0 = 1$? Using the division law, $a^m \div a^m = a^{m-m} = a^0$. But $\frac{a^m}{a^m} = 1$, so $a^0 = 1$ (for $a \neq 0$).

Related concepts

• Standard form and very large or small numbers
• Exponential functions and logarithms
• Understanding surds and simplifying radicals