Indices and the Index Laws
Indices, or powers, show repeated multiplication of a base number: a^n means a multiplied by itself n times. The index laws are shortcut rules for combining powers of the same base without expanding them in full: add the indices when multiplying, subtract when dividing, multiply the indices for a power of a power, any non-zero number to the power 0 equals 1, and a negative index means the reciprocal of the positive power.
Method
- Check that the bases are the same before applying any index law - the laws only work directly when every term shares the same base.
- To multiply powers of the same base, add the indices: a^m x a^n = a^(m+n).
- To divide powers of the same base, subtract the indices: a^m / a^n = a^(m-n).
- To raise a power to a further power, multiply the indices: (a^m)^n = a^(mn).
- Remember that any non-zero number raised to the power 0 equals 1, so a^0 = 1.
- A negative index means the reciprocal of the positive power: a^(-n) = 1/(a^n).
- Combine several laws in sequence for a multi-step expression, applying one rule at a time and simplifying as you go.
Worked example
Simplify (7^5 x 7^2) / 7^4, giving your answer as a single power of 7. Then work out its value.
- Apply the multiplication law to the numerator: 7^5 x 7^2 = 7^(5+2) = 7^7.
- Now divide by 7^4 using the division law: 7^7 / 7^4 = 7^(7-4) = 7^3.
- Evaluate 7^3 = 7 x 7 x 7 = 343.
- So (7^5 x 7^2) / 7^4 = 7^3 = 343.
Practice questions
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Q1Simplify 3^6 x 3^2, giving your answer as a single power of 3.Show answer
Answer: 3^8
Q2Simplify 8^9 / 8^5, giving your answer as a single power of 8.Show answer
Answer: 8^4
Q3Simplify (2^3)^4, giving your answer as a single power of 2. Then find its value.Show answer
Answer: 2^12 = 4096
Q4Write down the value of 15^0.Show answer
Answer: 1
Q5Write 6^-2 as a fraction.Show answer
Answer: 1/36
Q6Simplify (9^4 x 9^3) / 9^5, giving your answer as a single power of 9. Then find its value.Show answer
Answer: 9^2 = 81
Q7Work out the value of 2^-3.Show answer
Answer: 1/8
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Simplify (4^7 x 4^3) / 4^8, giving your answer as a single power of 4. Then work out its value.
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Write 3^-4 as a fraction in its simplest form. Then write it as a decimal, correct to 3 significant figures.
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A storage company can hold 2^10 gigabytes of data. Each year its capacity doubles. (a) Write an expression, as a single power of 2, for the amount it can store after 5 more years of doubling. (b) Work out this amount in gigabytes.
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Free printable worksheet
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This topic is chapter 12 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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