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Cambridge IGCSE0580

Indices I

Mathematics 0580 Chapter Notes

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Indices I
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In this chapter

  1. 1Positive and zero indices
  2. 2Negative indices
  3. 3The rules of indices
  4. 4Fractional indices
  5. 5Combining the rules

1. Positive and zero indices

An index (or power) says how many times a number is multiplied by itself. aⁿ means n copies of a multiplied together.

Any non-zero number to the power of 0 is 1. The base does not matter, so 143⁰ = 1 and 8⁰ = 1.

aⁿ = a × a × … × a (n times)

a⁰ = 1

Worked example 11 mark

8^(y) = 1

Find the value of y.

  1. 1

    1 = 8⁰, because any non-zero number to the power 0 is 1

  2. 2

    y = 0 [1]

2. Negative indices

A negative index means a reciprocal: 7⁻² = 1/7² = 1/49. It does not make the answer negative.

a⁻ⁿ = 1/aⁿ

Worked example 12 marks

Write 5⁻¹ as a decimal.

  1. 1

    5⁻¹ = 1/5 [1]

  2. 2

    = 0.2 [1]

3. The rules of indices

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

a^(m) × aⁿ = a^(m+n)

a^(m) ÷ aⁿ = a^(m-n)

(a^(m))ⁿ = a^(mn)

Examiner insight

“Candidates could improve by reviewing the rules for calculating with powers and the laws of indices.” “Adding the indices, instead of multiplying them was often seen.”

Worked example 12 marks

Write 9 × 27 as a power of 3.

  1. 1

    9 = 3² and 27 = 3³ [1]

  2. 2

    3² × 3³ = 3²⁺³ = 3⁵ [1]

4. Fractional indices

A fractional index is a root. The denominator is the root and the numerator is the power: 6^(1/2) = √6 and 16^(1/4) = √[4]16 = 2.

Take the root first, then the power. This keeps the numbers small on a non-calculator paper. The index is not a multiplier: 27^(2/3) is not 27 × 2/3.

A negative fractional index is the reciprocal of the answer you would get without the minus sign.

a^(1/n) = √[n]a

a^(()/mn) = (√[n]a)^(m)

Examiner insight

“The less successful candidates did not consider the most sensible strategy for a non-calculator paper, and attempted to square 125 first, then take the cube root of the result.” “Common errors included multiplying 27 by the fractional power.”

Worked example 13 marks

Do not use a calculator. Find the value of 8^(-2/3).

  1. 1

    8^(-2/3) = 1/(8^(2/3)) [1]

  2. 2

    8^(2/3) = (√[3]8)² = 2² = 4 [1]

  3. 3

    8^(-2/3) = 1/4 [1]

5. Combining the rules

Work out each part of a product separately, then combine the results.

Write a decimal index as a fraction: 0.5 = 1/2. With a negative fractional index there are three jobs: the root, the power, then the reciprocal. Do not stop after two.

Examiner insight

“Some candidates managed to apply the fractional power to 27 to reach 9 then did not deal with the negative to put the 9 in the denominator.”

Worked example 14 marks

Find the value of 27^(-2/3) × (6⁴)^(0.5).

  1. 1

    27^(-2/3) = 1/(27^(2/3)) = 1/((√[3]27)²) [1]

  2. 2

    = 1/3² = 1/9 [1]

  3. 3

    (6⁴)^(0.5) = 6^(4 × 0.5) = 6² = 36 [1]

  4. 4

    1/9 × 36 = 4 [1]

Recap

  • a⁰ = 1 for any non-zero a.
  • a⁻ⁿ = 1/aⁿ. A negative index gives a reciprocal, not a negative number.
  • Same base: multiply = add the indices, divide = subtract, power of a power = multiply.
  • Write numbers as powers of the same base (such as 9 = 3²) before using a rule.
  • Extended: a^(1/n) = √[n]a and a^(()/mn) = (√[n]a)^(m). Root first.
  • Extended: a negative fractional index needs the root, the power and then the reciprocal.

End-of-chapter exercise

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

  1. Do not use a calculator. Write down the value of 4³.1 mark
  2. Write 7⁻² as a fraction.2 marks
  3. Do not use a calculator. Find the value of 9⁰ + 2⁻³.2 marks
  4. Find the value of 2⁻³ × 2⁴.2 marks
  5. Do not use a calculator. Find the value of (2³)² ÷ 2⁴.2 marks
  6. 3⁷ ÷ 3^(w) = 3⁻² Find the value of w.2 marks
  7. Write (9 × 3⁴)/(3⁻²) as a single power of 3.3 marks
  8. Do not use a calculator. Find the value of 16^(3/4).2 marks
  9. Do not use a calculator. Here are four numbers. 4⁻² 8^(-2/3) 16^(-3/4) (2⁻³)² Write each number as a power of 2. Then write the four numbers in order, starting with the smallest.4 marks
  10. Do not use a calculator. (27^(-2/3) × 9^(k))/(3⁻¹) = 81^(3/4) Find the value of k.5 marks

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