Algebraic indices Revision Notes for AQA GCSE Maths

Algebraic indices

Algebraic indices are a way of writing repeated multiplication in a shorter form. When working with algebra, you'll often need to simplify expressions that contain powers, and understanding index laws will help you do this efficiently.

Understanding the parts

Before diving into the rules, it's important to know the terminology:

When you see an expression like $y^3$ , there are two key parts:

The base is the main letter or number (in this case, $y$ )
The index is the small number written above and to the right (in this case, $3$ )

The index tells you how many times to multiply the base by itself. For example, $n^5$ means $n \times n \times n \times n \times n$ .

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The words "index" and "power" mean the same thing in mathematics. You might hear both terms used interchangeably.

The four index laws

These laws help you simplify algebraic expressions containing powers. You can only use these laws when the bases are the same.

Law 1: Multiplying powers with the same base

When multiplying powers that have the same base, add the indices.

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

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

This works because you're essentially combining groups of the same base being multiplied together.

Law 2: Dividing powers with the same base

When dividing powers that have the same base, subtract the indices.

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

For example: $m^6 \div m^2 = m^{6-2} = m^5$

This makes sense because division cancels out some of the multiplied bases.

Law 3: Raising a power to another power

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

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

For example: $(r^2)^4 = r^{2 \times 4} = r^8$

This happens because you're repeating the original power multiple times.

Law 4: Powers inside brackets

When everything inside brackets is raised to a power, apply the power to each part inside the brackets.

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

For example: $(4b)^3 = 4^3 \times b^3 = 64b^3$

chatImportant

Remember: Same base only! You can only use these index laws when the bases are exactly the same. Different letters or numbers cannot be combined using these rules.

Step-by-step approach

For complex expressions with multiple operations, use the "one at a time" method:

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Systematic Problem-Solving Approach:

Deal with any number parts first - multiply or divide the numbers
Apply the index laws to work out the new power for each letter

This methodical approach prevents errors and ensures you don't miss any steps.

For example, with $7x \times 5x^6$ :

First: $7 \times 5 = 35$
Then: $x^1 \times x^6 = x^{1+6} = x^7$
Final answer: $35x^7$

Worked examples

Let's look at how to apply these laws in practice:

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Worked Examples: Applying Index Laws

Example 1: Simplify $p^2 \times p^7$

Same base $(p)$ , so add the indices: $2 + 7 = 9$
Answer: $p^9$

Example 2: Simplify $m^6 \div m^3$

Same base $(m)$ , so subtract the indices: $6 - 3 = 3$
Answer: $m^3$

Example 3: Simplify $(d^5)^3$

Power to a power, so multiply the indices: $5 \times 3 = 15$
Answer: $d^{15}$

Important restrictions

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Critical Rules to Remember:

You can only use these index laws when the bases are the same.

If there's no visible index, remember that the number has a power of $1$ . For example:

$n \times n^5 = n^{1+5} = n^6$
$x^2 \div x = x^{2-1} = x^1 = x$

You cannot use index laws with different bases - they must be identical letters or numbers.

Practice problems

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Try these problems to test your understanding:

Simplify $m^5 \times m^4$
Simplify $k^6 \div k$
Simplify $h^{11} \div h^3$
Simplify $(y^4)^2$
Simplify $(5p^2)^3$

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Key Points to Remember:

Index laws only work when bases are the same - you cannot combine different letters or numbers
Multiplication: add the indices ( $a^m \times a^n = a^{m+n}$ )
Division: subtract the indices ( $a^m \div a^n = a^{m-n}$ )
Power to a power: multiply the indices ( $(a^m)^n = a^{mn}$ )
Use "one at a time" for complex problems - deal with numbers first, then apply index laws to letters
Remember that no visible index means the power is 1

Algebraic indices (AQA GCSE Maths): Revision Notes