1MA1 · A4AlgebraFoundation & Higher

Index Laws

Index notation records repeated multiplication. The laws of indices let you simplify powers efficiently, but each law has conditions: most importantly, the base must be the same.

45 min Video by Maths Genie Pearson Edexcel GCSE Maths (1MA1)

The big picture

Indices compress repeated multiplication. For example, $x^{5}$ means five factors of $x$ ; it does not mean $5 x$ . Thinking in factors explains every index law and is safer than memorising a list of rules.

Edexcel questions mix several laws in one expression and may include numerical coefficients, zero, negative or fractional indices. A strong solution simplifies the numbers and the powers separately before combining them.

Indices (Laws of Indices)Watch the walkthrough, then read the notes below.

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What you'll learn

Use the product, quotient and power laws of indices
Simplify expressions containing coefficients and several variables
Interpret zero and negative indices
Evaluate fractional indices in Higher-tier questions

Multiplying and dividing powers

When powers with the same base are multiplied, add their indices: $a^{m} \times a^{n} = a^{m + n}$ . When they are divided, subtract: $a^{m} \div a^{n} = a^{m - n}$ , provided $a \neq = 0$ .

The base must match. You can simplify $x^{4} \times x^{3}$ , but not $x^{4} \times y^{3}$ . Deal with coefficients independently: $3 x^{2} \times 4 x^{5} = 12 x^{7}$ .

a^{m} \times a^{n} = a^{m + n}

Same base multiplied: add the indices.

a^{m} \div a^{n} = a^{m - n}

Same base divided: subtract the indices.

1Divide the coefficients:

18 \div 6 = 3

2Subtract the powers of

a

a^{7 - 2} = a^{5}

3Treat

b

b^{1}

b^{3 - 1} = b^{2}

Answer

3 a^{5} b^{2}

Why this works

The product law works because joining $m$ factors of $a$ to another $n$ factors creates $m + n$ factors. The quotient law cancels common factors, leaving the difference.

Powers of powers and brackets

Raising a power to another power multiplies the indices: $(a^{m})^{n} = a^{mn}$ . If a product is inside the brackets, every factor is raised: $(3 x^{2})^{3} = 3^{3} x^{6} = 27 x^{6}$ .

Brackets matter. $- 2^{4} = - (2^{4}) = - 16$ , while $(- 2)^{4} = 16$ because the negative sign is included in the repeated multiplication.

(a^{m})^{n} = a^{mn}

A power raised to a power: multiply the indices.

1Raise the coefficient:

2^{4} = 16

2Multiply the index on

p

p^{3 \times 4} = p^{12}

3Multiply the index on

q

q^{2 \times 4} = q^{8}

Answer

16 p^{12} q^{8}

Tip — A common lost mark is raising the letters but forgetting to raise the numerical coefficient.

Zero, negative and fractional indices

For any non-zero $a$ , $a^{0} = 1$ . A negative index means reciprocal, not a negative answer: $a^{- n} = 1/ a^{n}$ .

At Higher tier, a denominator in the index represents a root and a numerator represents a power: $a^{m / n} = n a^{m} = (n a)^{m}$ . Choose the order that keeps the arithmetic simple.

a^{0} = 1

Zero index, for

a \neq = 0

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

Negative index: take the reciprocal.

a^{m / n} = n a^{m}

Fractional index (Higher).

1The denominator 3 means take the cube root:

327 = 3

2The numerator 2 means square the result:

3^{2} = 9

Answer

9

Why this works

The zero and negative laws follow from the quotient law: $a^{3} / a^{3} = a^{3 - 3} = a^{0} = 1$ , and $a^{2} / a^{5} = a^{- 3} = 1/ a^{3}$ .

Think like an examiner

Check the operation before selecting a law: add indices for multiplication, subtract for division and multiply for a power of a power.
Leave answers with positive indices unless the question explicitly asks for another form.
In fractional-index evaluations, show the root and power stages so method marks remain available after an arithmetic slip.

Remember these

a^{m} a^{n} = a^{m + n}

Product law.

\frac{a ^{m}}{a ^{n}} = a^{m - n}

Quotient law.

(a^{m})^{n} = a^{mn}

Power law.

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

Negative index.

Watch out for these

x^{3} + x^{4} = x^{7}

The product law applies to multiplication, not addition. These unlike terms cannot be collected.

(3 x^{2})^{2} = 3 x^{4}

Square every factor:

(3 x^{2})^{2} = 9 x^{4}

2^{- 3} = - 8

A negative index creates a reciprocal:

2^{- 3} = 1/8

Stretch yourself

Simplify $\frac{( 6 x ^{4} y ^{- 2} ) ^{2}}{9 x ^{3} y ^{- 1}}$ and give the answer using positive indices.

Hint — Apply the outer power first, then divide coefficients and subtract the indices of matching bases.

Questions students ask

Why can I not use the product law when adding powers?

The law comes from joining repeated factors, which only happens during multiplication. For example,

x^{2} + x^{3}

is a sum and factorises to

x^{2} (1 + x)

Does a negative index make the value negative?

No. It moves the power across a fraction line:

a^{- n} = 1/ a^{n}

. The sign of the value still depends on the base.

Key takeaways

Index laws combine powers only when their bases match.
Multiplication adds indices, division subtracts them, and a power of a power multiplies them.
Zero gives 1, a negative index gives a reciprocal, and a fractional index represents a root and a power.

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