Powers & Roots
What are powers/indices?
- Powers of a number is when that number is multiplied by itself repeatedly
- 51 means 5
- 52 means 5 × 5
- 53 means 5 × 5 × 5
- Therefore the powers of 5 are 5, 25, 125, etc
- The big number on the bottom is sometimes called the base number
- The small number that is raised is called the index or the exponent
- Any non-zero number to the power of 0 is equal to 1
- 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
- The two square roots are 5 and -5
- Every positive number has two square roots
- They will have the same digits but one is positive and one is negative
- 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
- The square root of a negative number is not a real number
- The positive square root can be written as an index of so
- 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 so
- A nth root of a number ()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
- There will be a positive and negative nth root
- The notation refers to the positive nth root of a number
- If n is odd then they work the same way as cube
- Every positive and negative number will have an nth root
- The nth root can be written as an index of
- If n is even then they work the same way as square roots
- If you know your powers of numbers then you can use them to find roots of numbers
- e.g. means
- You could write this using an index
- e.g. means
- You can also estimate roots by finding the closest powers
- e.g. and therefore
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 0.25 or is 4
- The reciprocal of is
- 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