Powers, Roots & Indices (Edexcel GCSE Maths: Foundation)

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Jamie W

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Jamie W

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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 Tip

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

Laws of Indices

What are the laws of indices?

  • Index laws are rules you can use when doing operations with powers
    • They work with both numbers and algebra
Law Description How it works
a to the power of 1 equals a Anything to the power of 1 is itself 6 to the power of 1 equals 6
a to the power of 0 equals 1 Anything to the power of 0 is 1 8 to the power of 0 equals 1
a to the power of m cross times a to the power of n equals a to the power of m plus n end exponent To multiply indices with the same base, add their powers 4 cubed cross times 4 squared
equals open parentheses 4 cross times 4 cross times 4 close parentheses cross times open parentheses 4 cross times 4 close parentheses
equals 4 to the power of 5
a to the power of m divided by a to the power of n equals a to the power of m over a to the power of n equals a to the power of m minus n end exponent To divide indices with the same base, subtract their powers 7 to the power of 5 divided by 7 squared
equals fraction numerator 7 cross times 7 cross times 7 cross times up diagonal strike 7 cross times up diagonal strike 7 over denominator up diagonal strike 7 cross times up diagonal strike 7 end fraction
equals 7 to the power of 3 space end exponent
open parentheses a to the power of m close parentheses to the power of n equals a to the power of m n end exponent To raise indices to a new power, multiply their powers open parentheses 14 cubed close parentheses squared
equals open parentheses 14 cross times 14 cross times 14 close parentheses cross times open parentheses 14 cross times 14 cross times 14 close parentheses
equals 14 to the power of 6
open parentheses a b close parentheses to the power of n equals a to the power of n b to the power of n
To raise a product to a power, apply the power to both numbers, and multiply open parentheses 3 cross times 4 close parentheses squared equals 3 squared cross times 4 squared

a to the power of negative 1 end exponent equals 1 over a

a to the power of negative n end exponent equals 1 over a to the power of n

A negative power is the reciprocal

6 to the power of negative 1 end exponent equals 1 over 6

11 to the power of negative 3 end exponent equals 1 over 11 cubed

How do I deal with different bases?

  • Index laws only work with terms that have the same base
    • 2 cubed cross times 5 squared cannot be simplified using index laws
  • Sometimes expressions involve different base values, but one is related to the other by a power
    • e.g. 2 to the power of 5 cross times 4 cubed
  • You can use powers to rewrite one of the bases
    • 2 to the power of 5 cross times bold 4 cubed equals 2 to the power of 5 cross times open parentheses bold 2 to the power of bold 2 close parentheses cubed
    • This can then be simplified more easily, as the two bases are now the same
    • 2 to the power of 5 cross times open parentheses 2 squared close parentheses cubed equals 2 to the power of 5 cross times 2 to the power of 6 equals 2 to the power of 11

Worked example

(a)
Find the value of x when 6 to the power of 10 space cross times space 6 to the power of x space equals space 6 squared
 
Using the law of indices a to the power of m cross times a to the power of n equals a to the power of m plus n end exponent we can rewrite the left hand side
 
6 to the power of 10 cross times 6 to the power of x equals 6 to the power of 10 plus x end exponent
So the equation is now
 
6 to the power of 10 plus x end exponent equals 6 squared
 
Comparing both sides, the bases are the same, so we can say that
 
10 plus x equals 2
Subtract 10 from both sides
bold italic x bold equals bold minus bold 8
 
(b)
Find the value of n when 5 to the power of n divided by 5 to the power of 4 equals 5 to the power of 6
 
Using the law of indices a to the power of m divided by a to the power of n equals a to the power of m minus n end exponent we can rewrite the left hand side
 
5 to the power of n divided by 5 to the power of 4 equals 5 to the power of n minus 4 end exponent
So the equation is now 
5 to the power of n minus 4 end exponent equals 5 to the power of 6
 
Comparing both sides, the bases are the same, so we can say that
 
n minus 4 equals 6
Add 4 to both sides
bold italic n bold equals bold 10
 
(c)
Without using a calculator, find the value of 2 to the power of negative 4 end exponent
 
Using the law of indices a to the power of negative n end exponent equals 1 over a to the power of n we can rewrite the expression
 
2 to the power of negative 4 end exponent equals 1 over 2 to the power of 4
 
2 to the power of 4 equals 2 cross times 2 cross times 2 cross times 2 equals 16 so we can rewrite the expression
 
1 over 2 to the power of 4 equals 1 over 16
bold 1 over bold 16

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Jamie W

Author: Jamie W

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.