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Cambridge IGCSE0607

Indices I

International Mathematics 0607 Chapter Notes

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Indices I
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In this chapter

  1. 1Positive, zero and negative indices
  2. 2The rules of indices
  3. 3Fractional indices

1. Positive, zero and negative indices

An index says how many copies of a number are multiplied together: 5³ = 5 × 5 × 5 = 125.

A zero index gives 1. A negative index gives a reciprocal. It does not make the answer negative: 5⁻³ = 1/5³ = 1/125.

a⁰ = 1

a⁻ⁿ = 1/aⁿ

Examiner insight

“Some of the candidates who got the index correct in part (a) could not translate it into a fraction.” “A few candidates eliminated the negative power but did not evaluate the square index.”

Worked example 12 marks

Work out 5⁻³.

Give your answer as a fraction.

  1. 1

    5⁻³ = 1/5³ [1]

  2. 2

    = 1/125 [1]

2. The rules of indices

To multiply powers of the same base, add the indices. To divide, subtract the indices. To raise a power to a power, multiply the indices.

For example, 2⁻³ × 2⁴ = 2¹ = 2.

If the bases differ, first write them with the same base. To solve 2ⁿ = 64, write 64 as a power of 2.

a^(m) × aⁿ = a^(m+n)

a^(m) ÷ aⁿ = a^(m-n)

(a^(m))ⁿ = a^(mn)

Examiner insight

“Some candidates divided the indices instead of subtracting them.” “The majority of candidates scored full marks but a significant number did not realise that they had to write 4 as a power of 2 before simplification.”

Worked example 11 mark

2ⁿ = 64

Find the value of n.

  1. 1

    64 = 2 × 2 × 2 × 2 × 2 × 2 = 2⁶

  2. 2

    n = 6 [1]

3. Fractional indices

A fractional index means a root. The denominator says which root: 6^(1/2) is the square root of 6, and 16^(1/4) is the fourth root of 16.

With a numerator other than 1, take the root, then the power. A negative index still gives a reciprocal. For example, 81^(1/2) = 9 and 8^(-2/3) = 1/((√[3]8)²) = 1/4.

a^(1/n) = √[n]a

a^(m/n) = (√[n]a)^(m)

a^(-m/n) = 1/(a^(m/n))

Examiner insight

“Whilst many candidates gave the correct answer, there were a number of candidates who clearly did not know how to deal with a fractional index.”

Worked example 12 marks

Work out 125^(-2/3). Give your answer as a fraction.

  1. 1

    Root, then power: 125^(2/3) = (√[3]125)² = 5² = 25 [1]

  2. 2

    Negative index, so take the reciprocal: 125^(-2/3) = 1/25 [1]

Recap

  • Zero index: a⁰ = 1.
  • Negative index: a⁻ⁿ = 1/aⁿ. The answer is a reciprocal, not a negative number.
  • Same base, multiply: add the indices. Divide: subtract. Power of a power: multiply.
  • Different bases: write them with the same base first.
  • Extended: a^(1/n) is the nth root of a.
  • Extended: a^(m/n) = (√[n]a)^(m). Take the root, then the power.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Work out 4⁻². Give your answer as a fraction.2 marks
  2. Do not use a calculator. Work out 9⁰ + 3².2 marks
  3. Write 1/8 as a power of 2.2 marks
  4. Do not use a calculator. Write 3⁻² × 3⁵ as a single power of 3. Work out its value.2 marks
  5. Use your calculator to work out 7⁻². Give your answer as a decimal, correct to 3 significant figures.2 marks
  6. Do not use a calculator. Work out (2³)².2 marks
  7. Do not use a calculator. Write (4³ × 2⁻²)/2³ as a single power of 2. Work out its value.3 marks
  8. Do not use a calculator. Work out 16^(3/4).2 marks
  9. Do not use a calculator. Write (27^(2/3) × 9^(-1/2))/(81^(3/4)) as a single power of 3. Work out its value as a fraction.4 marks
  10. Do not use a calculator. In a model, a planet at a distance of d units from a star takes T years to orbit it and moves at a speed of v units, where T = d^(3/2) and v = d^(-1/2). Planet A has d = 16. Planet B has d = 1/4. Work out T for each planet. Then work out how many times faster planet B moves than planet A.5 marks

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