Algebraic indices Revision Notes for Edexcel GCSE Maths

Algebraic indices

What are algebraic indices?

Indices (singular: index) are small numbers written above and to the right of a base number or letter. They tell you how many times to multiply the base by itself.

The base is the main number or letter being multiplied
The index (or power) is the small number that shows how many times to multiply the base

For example: In $x^3$ , $x$ is the base and $3$ is the index. This means $x \times x \times x$ .

Index laws

These four rules help you simplify expressions with indices. Remember, you can only use these laws when the bases are the same.

Rule 1: Multiplying powers with the same base

When multiplying powers with the same base, add the indices.

Formula: $a^m \times a^n = a^{m+n}$

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Worked Example: Multiplying Powers

$x^4 \times x^3 = x^{4+3} = x^7$

This works because you're combining all the multiplications: $(x \times x \times x \times x) \times (x \times x \times x) = x^7$

Rule 2: Dividing powers with the same base

When dividing powers with the same base, subtract the indices.

Formula: $a^m \div a^n = a^{m-n}$

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Worked Example: Dividing Powers

$m^8 \div m^5 = m^{8-5} = m^3$

This works because you're cancelling out common factors from the top and bottom.

Rule 3: Raising a power to another power

When raising a power to another power, multiply the indices.

Formula: $(a^m)^n = a^{mn}$

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Worked Example: Power to a Power

$(j^2)^4 = j^{2 \times 4} = j^8$

This works because you're repeating the original multiplication four times.

Rule 4: Powers of products

When a product is raised to a power, each factor gets raised to that power.

Formula: $(ab)^n = a^n \times b^n$

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Worked Example: Powers of Products

$(4b)^3 = 4^3 \times b^3 = 64b^3$

This applies to everything inside the brackets.

Working with algebraic indices step by step

When simplifying expressions with both numbers and letters, follow this systematic approach:

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Step-by-Step Process:

Deal with numbers first - multiply or divide the numerical parts
Apply the index laws to the letter parts
Combine your results

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Worked Example: Mixed Numbers and Letters

Simplify: $7x \times 5x^6 = 35x^7$

Step 1: $7 \times 5 = 35$ Step 2: $x^1 \times x^6 = x^{1+6} = x^7$ Step 3: Combined result: $35x^7$

Important warnings

Same base rule

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You can only use index laws when the bases are exactly the same. If the bases are different, you cannot simplify further.

Correct: $x^2 \times x = x^3$ (same base: $x$ ) Incorrect: $x^2 \times y$ cannot be simplified (different bases)

Hidden index of 1

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When there's no visible index, the power is 1.

Example: $n \times n^5 = n^1 \times n^5 = n^6$

Brackets matter

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Powers only apply to what's directly attached unless there are brackets.

Example: $2x^3$ means $2 \times x^3$ , not $(2x)^3$

Practice examples

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Practice Problem 1: Multiplying

Simplifying $p^2 \times p^7$ : $p^2 \times p^7 = p^{2+7} = p^9$

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Practice Problem 2: Dividing

Simplifying $m^8 \div m^3$ : $m^8 \div m^3 = m^{8-3} = m^5$

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Practice Problem 3: Power to a Power

Simplifying $(a^6)^3$ : $(a^6)^3 = a^{6 \times 3} = a^{18}$

Remember!

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Key Points to Remember:

Add indices when multiplying powers with the same base
Subtract indices when dividing powers with the same base
Multiply indices when raising a power to another power
Index laws only work
when the bases are identical
No visible index means the power is 1

Algebraic indices (Edexcel GCSE Maths): Revision Notes