Powers & Roots (Cambridge (CIE) IGCSE International Maths)
Revision Note
Written by: Jamie Wood
Reviewed by: Dan Finlay
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 refers to the positive square root of a number
You can show both roots at once using the plus or minus symbol ±
Square roots of 25 are
What are cube roots?
A 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 refers to the cube root of a number
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 refers to the positive nth root of a number
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
We know that and
So 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
The reciprocal of is 4
The reciprocal of is
The reciprocal of a number can be written as an index of -1
5-1 is the reciprocal of 5, so
This can be extended to other negative indices
5-2 means the reciprocal of 52, so or
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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