Arithmetic : Exponents, Powers and Roots of Numbers

Powers

If a is any number, then

a² = a × a ;
a³ = a × a × a ; and
aⁿ = a × a × a × ... n times

where n is a positive integer.

a² is read as 'a squared', a³ is read as 'a cubed', and aⁿ is read as 'a raised to the nth power'.
Thus, the nth power of a or aⁿ is the number a used n times as a factor in a product.
For example, 6² = 6 × 6 = 36, (−5)³ = (−5) × (−5) × (−5) = −125, and 2⁶ = 2 × 2 × 2 × 2 × 2 × 2 = 64.

The following squares and cubes are worth noting:
1² = 1 ; 2² = 4 ; 3² = 9 ; 4² = 16 ; 5² = 25 ; 6² = 36 ; 7² = 49 ; 8² = 64 ; 9² = 81 ; 10² = 100 ; 11² = 121 ;
12² = 144 ; 13² = 169 ; 14² = 196 ; 15² = 225 ; 16² = 256 ; 17² = 289 ; 18² = 324 ; 19² = 361 ; 20² = 400 ;
1³ = 1 ; 2³ = 8 ; 3³ = 27 ; 4³ = 64 ; 5³ = 125 ; 6³ = 216 ; 7³ = 343 ; 8³ = 512 ; 9³ = 729 ; 10³ = 1000 ;

Squaring a number that is greater than 1, or raising it to a higher power, yields a larger number.
For example, the powers of 3 are 3² = 3 × 3 = 9, 3³ = 3 × 3 × 3 = 27, and 3⁴ = 3 × 3 × 3 × 3 = 81.
Squaring a number that is between 0 and 1, or raising it to a higher power, yields a smaller number.
For example, the powers of (1/3) are (1/3)² = 1/9, (1/3)³ = 1/27, and (1/3)⁴ = 1/81.
As another example, the powers of 0.1 are (0.1)² = 0.01, (0.1)³ = 0.001, and (0.1)⁴ = 0.0001.

Note that 0ⁿ = 0 and 1ⁿ = 1 for any positive integer n.
a⁰ = 1 for any non-zero number a, and 0⁰ is not defined.
a¹ = a (i.e., if the power is 1, it is understood and usually not written).
a² ≥ 0 for any a.
The square is always non-negative because the product of two negative numbers is positive.

If a fraction (a/b) is raised to the nth power, then (a/b)ⁿ = aⁿ/bⁿ.

Example
If the value of an investment in the stock market increases by 25% each year, what will be the value of a $8000 investment in 3 years?
Solution.

Each year, the investment increases by 25%, i.e., a factor of 1.25
In 3 years, the investment increases by a factor of 1.25 × 1.25 × 1.25 = (1.25)³
(1.25)³ = (5/4)³ = 5³/4³ = 125/64
$8000 × 125/64 = 1000 × 125/8 = 1000 × 15.625 = 15625
The value of the investment in 3 years will be 15625.