Powers & Roots (Cambridge International IGCSE Mathematics)

Powers & roots

What are powers (indices)?

• Powers (or indices) are the small 'floating' values that are used when a number is multiplied by itself repeatedly

  • 6¹ means 6
  • 6² means 6 × 6
  • 6³ means 6 × 6 × 6

• The big number at the bottom is called the base

• The small number that is raised is called the index, power, or exponent

• Any non-zero number to the power of 0 is equal to 1

  • 3⁰ = 1

• Any number to the power of 1 is equal to itself

  • 3¹=3

What are square roots?

• Roots are the reverse of powers

• A square root of 25 is a number that when squared equals 25

  • The two square roots of 25 are 5 and -5

    • 5² = 25 and (-5)² = 25

• Every positive number has two square roots

  • One is positive and one is negative
  • Negative numbers do not have a real square root

• The notation $\sqrt{}$ refers to the positive square root of a number

  • $\sqrt{25}=5$
  • You can show both roots at once using the plus or minus symbol±
  • Square roots of 25 are $\pm\sqrt{25}=\pm5$

What are cube roots?

• A cube root of 125 is a number that when cubed equals 125

  • The cube root of 125 is 5

    • 5³ = 125

  • Unlike square roots, each number only has one cube root

  • Every positive and negative number has one real cube root

  • The notation $\sqrt[3]{ }$ refers to the cube root of a number

    • $\sqrt[3]{125} = 5$

What are n^th roots?

• An nth root of a number is a value that when raised to the power n equals the original number

  • 3⁵=243 therefore 3 is a 5th root of 243

• If n is even, there will be a positive and negative nth root

  • The 6th roots of 64 are 2 and -2

    • 2⁶ = 64 and (-2)⁶ = 64

  • The notation n-th root of blank refers to the positive nth root of a number

    • root index 6 of 64 equals 2

  • Negative numbers do not have a real nth root if n is even

• If n is odd then there will only be one n*th root

  • The 5th root of -32 is -2

    • (-2)⁵ = -32equation

  • Every positive and negative number will have one real nth root

How do I estimate a root?

• You can estimate roots by finding the closest integer roots

  • To estimate $\sqrt{20}$

    • We know that $\sqrt{16}=4$ and $\sqrt{25}=5$
    • So $\sqrt{20}$ must be between 4 and 5

What are reciprocals?

• The reciprocal of a number is 1 divided by the number

  • The reciprocal of 2 is $\frac{1}{2}$

• A number and its reciprocal multiply together to give 1

• You switch the numerator and denominator of a fraction to find its reciprocal

  • The reciprocal of $\frac{1}{4}$ is 4
  • The reciprocal of $\frac{3}{2}$ is $\frac{2}{3}$

• The reciprocal of a number can be written as an index of -1

  • 5^-1 is the reciprocal of 5, so $\frac{1}{5}$

• This can be extended to other negative indices

  • 5^-2 means the reciprocal of 5², so $\frac{1}{5^{2}}$ or $\frac{1}{25}$

Exam Hint If your calculator shows "Math Error" or similar when finding a square root, this is probably because you have accidentally entered a negative number!