Lesson 1.3 · 30 min

Index Laws and Negative Indices

The four index laws, what a negative index really means, and using the order of operations with powers.

In this lesson you will learn to…

Use the multiplication, division, zero and power laws of indices
Explain why $a^{- n} = \frac{1}{a ^{n}}$
Write numbers such as $\frac{1}{16}$ and $0.001$ as a power
Simplify expressions with several index laws, working in the right order

You already know that $2^3$ means $2 \times 2 \times 2$ . But what could $2^{-3}$ possibly mean? You can’t multiply 2 by itself minus three times. Instead of being told, work it out:

Investigate

Start at

2^{3} = 8

. Each step right, the power goes down by 1 — and the value halves. Keep the pattern going past zero.

Power	$2^{3}$	$2^{2}$	$2^{1}$	$2^{0}$	$2^{- 1}$	$2^{- 2}$	$2^{- 3}$
Value	8	Row 1, 22	Row 1, 21	Row 1, 20	Row 1, 2−1	Row 1, 2−2	Row 1, 2−3

The pattern decides

Nobody invented negative powers by decree — the pattern forced them. Every step to the right divides by 2 and lowers the power by 1, and there is no reason for that to stop at $2^0$ . So

$a^{-n} = \frac{1}{a^n}$

and in particular $a^0 = 1$ for any number except 0. We call $a^{-n}$ the reciprocal of $a^n$ .

The same pattern works in tens, which is why you will see these all the time in standard form: $10^{-1} = 0.1$ , $10^{-2} = 0.01$ , $10^{-3} = 0.001$ .

The laws — and the catch

Write $a^5 \times a^3$ out in full and you get eight $a$ s multiplied together, so the powers add. Dividing cancels $a$ s from top and bottom, so the powers subtract. And $(a^5)^3$ is three lots of $a^5$ , so the powers multiply.

The catch: all of that only works when the base is the same.

Several laws in one go

Order of operations still rules — Brackets, Indices, Division/Multiplication, Addition/Subtraction — and a long fraction line counts as brackets.

Key rules and formulas

Multiplying — add the powers
$a^{x} \times a^{y} = a^{x + y}$
Only for the same base.
Dividing — subtract the powers
$a^{x} \div a^{y} = a^{x - y}$
The answer can be negative — that is fine.
Power of a power — multiply
$(a^{x})^{y} = a^{x y}$
$(3^{10})^{2} = 3^{10} \times 3^{10} = 3^{20}$ .
Zero and negative indices
$a^{0} = 1 a^{- n} = \frac{1}{a ^{n}}$
A negative index means one over. It never makes the number negative.

Worked examples

Worked example 1

Two ways to see a negative power

Work out

6^{3} \div 6^{5}

in two ways.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Several laws at once

Simplify

\frac{( 3 ^{6} \times 3 ^{4} ) ^{2}}{3 ^{2} \times 3 ^{3}}

Try it yourself first, then reveal the solution one step at a time.

Practice: index laws

11 questions

Eleven questions on the index laws. Where a box asks for a power, type just the number — for example -3.

Question 1 of 11Flashcards
Flip through the index laws. Say each one out loud before you turn the card over.
Card 1 of 7
Tap the card to flip it
Question 2 of 11Type the answerFoundation
Write $3^{- 2}$ as a fraction.
Your answer
Question 3 of 11Fill the gaps
Write each number as a power of 2 by filling in the missing index.
$8 = 2^{□}$ : Gap a $2 = 2^{□}$ : Gap b $\frac{1}{2} = 2^{□}$ : Gap c $\frac{1}{16} = 2^{□}$ : Gap d $1 = 2^{□}$ : Gap e
Question 4 of 11Match the pairs
Match each power of 3 to its value.
Pick an answer, then choose the row it belongs to. You can also drag it.
- $3^{3}$
- $3^{- 2}$
- $3^{- 1}$
- $3^{0}$
- $3^{- 4}$
Question 5 of 11Type the answer
Work out $16 \times 2^{- 3}$ .
Your answer
Question 6 of 11Choose all that applyChallenge
Select every statement that is true.
Select every correct answer — there may be more than one.
$(- 2)^{- 1} < 0.5$ $6^{- 3} < 0$ $2^{- 3} > 2^{- 2}$ $5^{0} > 2^{0}$ $0 > - 3^{2}$
Question 7 of 11Fill the gaps
Simplify each expression, leaving your answer in index form. Type only the missing power.
$5^{- 7} \times 5^{5} = 5^{□}$ : Gap a $2^{- 1} \div 2^{5} = 2^{□}$ : Gap b $(6^{3})^{- 2} = 6^{□}$ : Gap c $1 1^{- 8} \times 1 1^{2} \times 1 1^{3} = 1 1^{□}$ : Gap d
Question 8 of 11Put in order
Put the steps for simplifying $\frac{( 3 ^{6} \times 3 ^{4} ) ^{2}}{3 ^{2} \times 3 ^{3}}$ in the correct order.
Put these in the right order. Drag them, or use the arrow buttons.
1. Divide: $3^{20} \div 3^{5} = 3^{15}$
2. Simplify the bottom: $3^{2} \times 3^{3} = 3^{5}$
3. Apply the power: $(3^{10})^{2} = 3^{20}$
4. Inside the top bracket: $3^{6} \times 3^{4} = 3^{10}$
Question 9 of 11Algebraic answer
Simplify $x^{3} \times x^{- 5}$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.
Question 10 of 11Step by stepChallenge
Simplify this one step at a time.
Simplify $\frac{( 2 ^{4} \times 2 ^{- 1} ) ^{5}}{2 ^{6} \times 2 ^{5}}$ and give the answer as an ordinary number.
1. $2^{4} \times 2^{- 1} = 2^{□}$ . What is the power?
  Step 1 answer
Question 11 of 11Quick-fire drill
Quick-fire: negative and zero powers. Give fractions like 1/8.
8 questions · 60 seconds · answer as many as you can.

In this lesson you will learn to…

The pattern decides

The laws — and the catch

Several laws in one go

Key rules and formulas

Multiplying — add the powers

Dividing — subtract the powers

Power of a power — multiply

Zero and negative indices

Worked examples

Practice: index laws