Squares, Cubes & Roots (Cambridge International IGCSE Mathematics)

Squares, cubes & roots

What are square numbers?

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

  • The first square number is $1\times1=1$, the second is $2\times2=4$ 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

  • $a\times a=a^{2}$

What are cube numbers?

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

  • The first cube number is $1\times1\times1=1$, the second is $2\times2\times2=8$ 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

  • $a\times a\times a=a^{3}$

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 inverse of squaring

  • Square roots are indicated by the symbol $\sqrt{}$

    • e.g. The positive square root of 49 would be written as $\sqrt{49}$

  • 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. $\sqrt{25}=5$ but $-\sqrt{25}=-5$
    • Sometimes both positive and negative square roots are of interest and would be indicated by $\pm\sqrt{25}$

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

  • e.g. $\sqrt{3}$ is a surd, as 3 is not a square number

  • surds are irrational numbers

    • where possible modern calculators will display irrational numbers as surds

  • $\sqrt{64}$ is rational, as it is equal to 8

    • 64 is a square number

  • However, $\sqrt{2}$ is irrational

    • 2 is not a square number

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

  • $\sqrt{1},\sqrt{4},\sqrt{9},\sqrt{16},\sqrt{25},\sqrt{36},\sqrt{49},\sqrt{64},\sqrt{81},\sqrt{100},\sqrt{121},\sqrt{144},\sqrt{169},\sqrt{196},\sqrt{225}$

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 inverse of cubing

  • Cube roots are indicated by the symbol $\sqrt[3]{ }$

    • e.g. The cube root of 64 would be written as $\sqrt[3]{64}$

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

    • ∛(1 comma ∛(8 comma ∛(27 comma ∛(64 comma ∛(125 comma ∛(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.

Answer:

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 8^thsquare number and 4^th cube number

Final answer: 64 is both a square and cube number 64 = 8²and 64 =4³