Powers, Roots & Indices | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Powers & Roots

What are powers/indices? 

• Powers of a number involve multiplying that number by itself repeatedly. 
• For example,
  5¹ equals 5
  5² equals 5 × 5
  5³ equals 5 × 5 × 5
• The base number is the larger number at the bottom, while the index or exponent is the smaller number raised.
• Any non-zero number to the power of 0 equals 1, as in 5⁰ = 1.

What are roots?

• Roots of a number are the opposite of powers.
• A square root of 25 is a number that, when squared, equals 25. For instance, the square roots of 25 are 5 and -5. Every positive number has two square roots - one positive and one negative.
  • The notation √ refers to the positive square root of a number.
    • √25 = 5
    • To represent both roots simultaneously, we can use the ± symbol.
    • Square roots of 25 are ±√25 equals ± 5
  • The square root of a negative number is not a real number
  • The positive square root can be written as an index of
• A cube root of 125 is a number that when cubed equals 125
  • A cube root of 125 is 5
  • Every positive and negative number always has a cube root
  • The notation∛ refers to the cube root of a number
    • 
  • The cube root can be written as an index of
  • A nth root of a number (n√)is a number that when raised to the power n equals the original number
    • If n is even then they work the same way as square roots
      • Every positive number will have a positive and negative nth root
      • The notation n√ refers to the positive nth root of a number
    • If n is odd then they work the same way as cube roots
      • Every positive and negative number will have an nth root
    • The nth root can be written as an index of 1/n
  • If you know your powers of numbers then you can use them to find roots of numbers
  • e.g. 2⁵ equals 32 means 5√ 32 = 2

    • You could write this using an index

    • You can also estimate roots by finding the closest powers
      

What are reciprocals?

• The reciprocal of a number is the number that you multiply it by to get 1
  • The reciprocal of 2 is 1/2
  • The reciprocal of 0.25 or 14
  • The reciprocal of 3/2 is 2/3
• The reciprocal of a number can be written as a power with an index of -1
  • 5^-1 means the reciprocal of 5
• This idea can be extended to other negative indices
  • 5^-2 means the reciprocal of 5²

Laws of Indices

What are the laws of indices?

• Numerous fundamental laws or rules exist.
• It's crucial to comprehend and effectively apply these rules.
• Grasping the explanations behind these rules aids in their retention and application.

How do I apply more than one of the laws of indices?

• Powers can include negatives and fractions
  • These can be dealt with in any order
  • However the following order is easiest as it avoids large numbers
• If there is a negative sign in the power then deal with that first
  • Take the reciprocal of the base number
  • 
• Next deal with the denominator of the fraction of the power
  • Take the root of the base number
  • 
• Finally deal with the numerator of the fraction of the power
  • Take the power of the base number
  • 

How do I deal with different bases?

• Occasionally, expressions feature various base values.
• Index laws can be utilized to convert the base of a term, simplifying an expression that incorporates terms with differing bases.
• For example
• Using the above can then help with problems like