Squares, Cubes & Roots (Cambridge (CIE) IGCSE Maths): Revision Note

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

Squares, Cubes & Roots

What are square numbers?

  • A square number is the result of multiplying a number by itself

    • The first square number is 1 cross times 1 equals 1, the second is 2 cross times 2 equals 4 and so on

  • The first 15 square numbers are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

    • Aim to remember at least the first fifteen square numbers

  • In algebra, square numbers can be written using a power of 2

    • a cross times a equals a squared

What are cube numbers?

  • A cube number is the result of multiplying a number by itself, twice

    • The first cube number is 1 cross times 1 cross times 1 equals 1, the second is 2 cross times 2 cross times 2 equals 8 and so on

  • The first 5 cube numbers are 1, 8, 27, 64 and 125

    • Aim to remember at least the first five cube numbers

    • You should also remember 103 = 1000

  • In algebra, cube numbers can be written using a power of 3

    • a cross times a cross times a equals a cubed

What are square roots?

  • The square root of a value, is the number that when multiplied by itself equals that value

    • For example, 4 is the square root of 16 

    • It is the opposite of squaring

    • Square roots are indicated by the symbol square root of space

      • e.g.  The square root of 49 would be written as square root of 49

    • Square roots can be positive and negative

      • e.g.  The square roots of 25 are 5 and -5

    • If a negative square root is required then a - sign would be used

      • e.g.  square root of 25 equals 5 but negative square root of 25 equals negative 5

      • Sometimes both positive and negative square roots are of interest and would be indicated by plus-or-minus square root of 25

  • The square root of a non-square integer is also called a surd

    • e.g. square root of 3 is a surd, as 3 is not a square number

    • surds are irrational numbers

      • where possible modern calculators will display irrational numbers as surds

    • square root of 64 is rational, as it is equal to 8

      • 64 is a square number

    • However, square root of 2 is irrational

      • 2 is not a square number

  • You should aim to remember the square roots of the first 15 square numbers:

    • square root of 1 comma space square root of 4 comma space square root of 9 comma space square root of 16 comma space square root of 25 comma space square root of 36 comma space square root of 49 comma space square root of 64 comma space square root of 81 comma space square root of 100 comma space square root of 121 comma space square root of 144 comma space square root of 169 comma space square root of 196 comma space square root of 225

What are cube roots?

  • The cube root of a value, is the number that when multiplied by itself twice equals that value

    • For example, 3 is the cube root of 27

    • It is the opposite of cubing

    • Cube roots are indicated by the symbol cube root of space space end root

      • e.g.  The cube root of 64 would be written as cube root of 64

    • You should remember the values of the following cube roots:

      • cube root of 1 comma space cube root of 8 comma space cube root of 27 comma space cube root of 64 comma space cube root of 125 comma space cube root of 1000

Worked Example

Write down a number which is both a cube number and a square number, and hence express this number in two different ways using powers of 2 and 3.

Listing the first 12 square numbers

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

Listing the first 5 cube numbers

1, 8, 27, 64, 125

64 appears in both lists, it is the 8th square number and 4th cube number

64 is both a square and cube number
64 = 82 and 64 =43

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Jamie Wood

Author: Jamie Wood

Expertise: Maths

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.