Indices
Indices is a key number topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on applying the laws of indices, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. When multiplying powers you add the indices; when dividing you subtract them.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
Topic overview
Indices, also called powers, are a shorthand for repeated multiplication. In \(3^4\) the base is \(3\) and the index is \(4\), meaning four threes multiplied together.
Three laws do most of the work. Multiplying powers of the same base adds the indices, dividing subtracts them, and raising a power to another power multiplies them. These only apply when the bases match, which is the condition students most often forget.
Two special cases follow from the laws. Anything to the power \(0\) equals \(1\), and a negative index means the reciprocal. Both look strange at first but they fall out naturally from the division rule.
Revision notes
The three laws
When multiplying, add the indices: \(a^m \times a^n = a^{m+n}\). When dividing, subtract them: \(a^m \div a^n = a^{m-n}\). For a power of a power, multiply: \((a^m)^n = a^{mn}\).
So \(2^3 \times 2^4 = 2^7\), \(5^6 \div 5^2 = 5^4\), and \((3^2)^5 = 3^{10}\). All three require the same base.
Coefficients and bases
A number in front of the power is handled separately. In \(3x^2 \times 4x^5\), multiply the coefficients to get \(12\) and add the indices to get \(x^7\), giving \(12x^7\).
The laws never apply across different bases, so \(2^3 \times 3^2\) cannot be simplified into a single power.
Zero and negative indices
Dividing \(a^3\) by \(a^3\) gives \(a^0\), and any number divided by itself is \(1\), so \(a^0 = 1\).
Continuing the pattern downwards, \(a^{-n} = \frac{1}{a^n}\). So \(2^{-3} = \frac{1}{8}\), which is a small positive number, not a negative one.
Key points
- \(a^m \times a^n = a^{m+n}\).
- \(a^m \div a^n = a^{m-n}\).
- \((a^m)^n = a^{mn}\).
- The laws apply only when the bases are the same.
- \(a^0 = 1\) for any non-zero \(a\).
- \(a^{-n} = \frac{1}{a^n}\).
Worked examples
Example 1
Simplify \(x^5 \times x^3\).
Working
Example 2
Simplify \(\frac{12x^7}{3x^2}\).
Working
Example 3
Work out \(2^{-3}\).
Working
Common mistakes
Multiplying the indices when multiplying powers.
2³ × 2⁴ is 2⁷, not 2¹². Multiplying the bases adds the indices.
Applying the laws across different bases.
2³ × 3² cannot be simplified into one power, because the bases differ.
Thinking a negative index gives a negative answer.
2⁻³ is 1/8, a small positive number. The negative refers to the reciprocal, not the sign.
Forgetting the coefficient.
In 3x² × 4x⁵ the numbers multiply to 12 while the indices add, giving 12x⁷.
Exam tips
- Deal with the coefficients and the powers as two separate steps.
- Check the bases match before applying any law.
- Rewrite negative indices as fractions before evaluating.
- Remember that anything to the power zero is 1.
Key terms
- Base
- The number being raised to a power, such as the 3 in \(3^4\).
- Index
- The power itself, showing how many times the base multiplies.
- Coefficient
- The number in front of a term, such as the 5 in \(5x^2\).
- Reciprocal
- One divided by a number, which is what a negative index produces.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.