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
- The cube root of 125 is 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 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
- Every positive and negative number will have an nth root
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
- To estimate
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 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!