Powers & Roots (Cambridge (CIE) IGCSE International Maths)

Revision Note

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 are 5 and -5 

  • Every positive number has two square roots

    • One is positive and one is negative

    • Negative numbers do not have a square root

  • The notation square root of blank end root  refers to the positive square root of a number

    • square root of 25 equals 5

    • You can show both roots at once using the plus or minus symbol ±

    • Square roots of 25 are plus-or-minus square root of 25 equals plus-or-minus 5

What are cube roots?

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

    • The cube root of 125 is 5

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

    • Every positive and negative number has a cube root

    • The notation cube root of blank refers to the cube root of a number

      • cube root of 125 equals 5

What are nth 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

    • 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 an nth root if n is even

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

    • Every positive and negative number will have an nth root

      • The 5th root of -32 is -2

How do I estimate a root?

  • You can estimate roots by finding the closest integer roots

    • To estimate square root of 20

      • We know that square root of 16 equals 4 and square root of 25 equals 5

      • So square root of 20 must be between 4 and 5

What are reciprocals?

  • The reciprocal of a number is the number that you multiply it by to get 1

    • The reciprocal of 2 is 1 half

    • The reciprocal of 1 fourth is 4

    • The reciprocal of 3 over 2 is 2 over 3

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

    • 5-1 is the reciprocal of 5, so 1 fifth

  • This can be extended to other negative indices

    • 5-2 means the reciprocal of 52, so 1 over 5 squared or 1 over 25

Examiner Tips and Tricks

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

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Jamie Wood

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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.

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