Powers, Roots and Standard Form

Number

A power, also called an index or exponent, is shorthand for repeated multiplication of one value by itself
- 7² means 7 × 7
- 2⁴ means 2 × 2 × 2 × 2
The number being multiplied is the base, and the raised number is the index
Any number raised to the power 1 is unchanged
Any non-zero number raised to the power 0 equals 1

Taking a square root reverses squaring
Squaring destroys sign, so two different values square to the same positive result
- 9 and −9 both square to 81, and each therefore counts as a square root of 81
The symbol √ denotes the positive square root only
- √81 = 9
Writing ± in front of the symbol calls for both roots
- ±√81 = ±9
No negative number has a real square root, since no real number squares to give a negative
A root that is not exact can be trapped between the two nearest whole roots
- 16 and 25 are the square numbers either side of 20, so √20 lies between 4 and 5

A cube root reverses cubing, and every number has exactly one real cube root
- ∛216 = 6, and ∛(−216) = −6
Cubing preserves sign, which is why negatives cause no difficulty here
An nth root is the value that, raised to the power n, returns the original number
An even value of n behaves like the square root: two real roots for a positive number, none for a negative
An odd value of n behaves like the cube root: exactly one real root, of the same sign as the number