Skip to content
Cambridge IGCSE0607

Indices II

International Mathematics 0607 Chapter Notes

What this chapter covers

Indices II
ShareWhatsAppPost
Indices II notes

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Indices II notes as text: skim, search, and jump between subtopics.

~5 min read

In this chapter

  1. 1Positive, zero and negative indices
  2. 2Multiplying and dividing powers
  3. 3A power of a power
  4. 4Fractional indices
  5. 5Fractional indices in algebra
  6. 6Equations with an unknown index
  7. 7Indices in a formula

1. Positive, zero and negative indices

An index (or power) says how many of the base are multiplied together. 2⁴ = 2 × 2 × 2 × 2 = 16.

To find an unknown index, count the factors. 32 = 2 × 2 × 2 × 2 × 2 = 2⁵, so if 2^(x) = 32 then x = 5.

Any number to the power 0 is 1. So 7⁰ = 1, not 7 and not 0.

A negative index means a reciprocal. The sign of the index does not make the answer negative. After taking the reciprocal, still work out the power: 4⁻² = 1/4² = 1/16.

a⁰ = 1

a⁻ⁿ = 1/aⁿ

Examiner insight

“A few candidates eliminated the negative power but did not evaluate the square index.” “A common error was not dealing with the negative index and giving a final answer of 5.”

Worked example 12 marks

Do not use a calculator. Find the value of 5⁻² + 7⁰.

  1. 1

    5⁻² = 1/5² = 1/25 and 7⁰ = 1 [1]

  2. 2

    1/25 + 1 = 11/25 [1]

2. Multiplying and dividing powers

To multiply powers of the same base, add the indices. To divide, subtract the indices.

With a coefficient, treat the numbers and the indices separately: divide the numbers, then subtract the indices. Subtracting a negative index adds it.

With two letters, deal with each letter on its own: 6x⁷y⁴ × 5x⁻⁵y = 30x²y⁵.

A letter with no index has index 1, so y = y¹.

a^(m) × aⁿ = a^(m+n)

a^(m) ÷ aⁿ = a^(m-n)

Examiner insight

“Some candidates multiplied the indices instead of adding them.” “The common errors occurred where candidates did not use different operations on the number and the indices.”

Worked example 12 marks

Simplify. 6x⁵ ÷ 3x⁻²

  1. 1

    6 ÷ 3 = 2 and x⁵ ÷ x⁻² = x⁵⁻⁽⁻²⁾ [1]

  2. 2

    = 2x⁷ [1]

3. A power of a power

When a bracket is raised to a power, multiply the indices: (2³)⁴ = 2¹². Adding them gives the wrong index, 7.

Every part of the bracket takes the power, the number as well.

In a fraction, the top and the bottom both take the power: (2x/3)³ = 8x³/27.

(a^(m))ⁿ = a^(mn)

(ab)ⁿ = aⁿ bⁿ

Examiner insight

“An answer of 7 came from a confusion of the rules of indices as, in this situation, the indices multiply not add.”

Worked example 12 marks

Do not use a calculator. Simplify (5x³)².

  1. 1

    (5x³)² = 5² × (x³)² [1]

  2. 2

    = 25x⁶ [1]

4. Fractional indices

A fractional index is a root. The denominator is the root and the numerator is the power.

Take the root first, then the power. The numbers stay small: 16^(3/4) = (√[4]16)³ = 2³ = 8.

A negative fractional index also means a reciprocal.

a^(1/n) = √[n]a

a^(m/n) = (√[n]a)^(m)

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

Examiner insight

“There were a number of candidates who clearly did not know how to deal with a fractional index.”

Worked example 12 marks

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

  1. 1

    27^(1/3) = 3, so 27^(-2/3) = 3⁻² [1]

  2. 2

    = 1/9 [1]

5. Fractional indices in algebra

The rules work on every part of a term. Raise the number to the power, and multiply the index of each letter by the power.

A fractional power of a bracket takes the root of the number and multiplies the index of each letter by the fraction.

Fractional indices add and subtract like whole ones: 3x^(-3/4) × 3x^(1/2) = 9x^(-1/4).

Examiner insight

“However, many candidates seemed to be more confident dealing with the algebraic indices rather than the purely numerical one.” “There were also a number of candidates who reached the correct answer and then either spoilt it or gave a less simplified one than they had already found.”

Worked example 14 marks

Simplify.

(a) (3y³)⁴

(b) (16x¹⁶)^(1/4)

  1. 1

    (a) (3y³)⁴ = 3⁴ × y^(3 × 4) [1]

  2. 2

    = 81y¹² [1]

  3. 3

    (b) 16^(1/4) = 2 and (x¹⁶)^(1/4) = x⁴ [1]

  4. 4

    = 2x⁴ [1]

6. Equations with an unknown index

To solve an equation with the unknown in the index, write both sides as a power of the same base. Then the indices are equal, and you solve for the unknown.

Write each number as a power of the smallest base that fits, for example 4 = 2². In 32^(x) = 2, write 32 = 2⁵. Then 2^(5x) = 2¹, so 5x = 1 and x = 1/5.

Examiner insight

“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 12 marks

3¹² ÷ 3ⁿ = 3²ⁿ

Find the value of n.

  1. 1

    3¹² ÷ 3ⁿ = 3¹²⁻ⁿ, so 12 - n = 2n [1]

  2. 2

    12 = 3n, so n = 4 [1]

7. Indices in a formula

A question can put indices inside a formula. Find the length first by taking a root, then use the formula.

A cube of volume V has edge V^(1/3), and its surface area is six faces.

V = l³

surface area = 6l²

Worked example 13 marks

The volume of a cube is 729x⁶ cm³. Find an expression, in terms of x, for the total surface area of this cube.

  1. 1

    edge = (729x⁶)^(1/3) = 9x² cm [1]

  2. 2

    one face = (9x²)² = 81x⁴ cm² [1]

  3. 3

    total = 6 × 81x⁴ = 486x⁴ cm² [1]

Recap

  • Index laws: a^(m) × aⁿ = a^(m+n), a^(m) ÷ aⁿ = a^(m-n), (a^(m))ⁿ = a^(mn).
  • a⁰ = 1 for any number a.
  • a⁻ⁿ = 1/aⁿ. A negative index gives a reciprocal, not a negative number.
  • With coefficients, divide or multiply the numbers, and add or subtract the indices.
  • a^(1/n) is the nth root of a. a^(m/n) is the root first, then the power.
  • To solve a^(x) = a^(y) type equations, write both sides with the same base and equate the indices.
  • Calculator papers: use the power key and put brackets round a negative or fractional index.

End-of-chapter exercise

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

  1. Write down the value of 9⁰.1 mark
  2. Do not use a calculator. Find the value of 2⁻⁴.1 mark
  3. Simplify 4x³ × 5x⁴.2 marks
  4. Do not use a calculator. Simplify 12a⁵ ÷ 3a⁻².2 marks
  5. Use a calculator to find the value of 200^(-1/3), correct to 3 significant figures.2 marks
  6. Do not use a calculator. Solve 2^(x) = 1/8.1 mark
  7. Do not use a calculator. Find the value of 64^(2/3).2 marks
  8. Do not use a calculator. Find the value of (4/9)^(-3/2).3 marks
  9. Do not use a calculator. A cuboid has a square base. Each side of the base is (16x⁶)^(1/4) cm. The volume of the cuboid is 12x^(7/2) cm³. Find an expression, in terms of x, for the height of the cuboid. Give your answer in its simplest form.4 marks
  10. Do not use a calculator. A rectangle has length 4^(x+1) cm and width 8^(-x) cm. Its area is 1/(√2) cm². Find the value of x.4 marks

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Types of numberSetsPowers and rootsFractions, decimals and percentagesOrderingThe four operations