Arithmetic : Exponents, Powers and Roots of Numbers

Preparation

Just what you need to know !

Powers

If a is any number, then

a^{2} = a × a ; a^{3} = a × a × a ; and a^{n} = a × a × a × ... n times

where n is a positive integer.

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

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 3^{2} = 3 × 3 = 9, 3^{3} = 3 × 3 × 3 = 27, and 3^{4} = 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 0^{n} = 0 and 1^{n} = 1 for any positive integer n. a^{0} = 1 for any non-zero number a, and 0^{0} is not defined. a^{1} = a (i.e., if the power is 1, it is understood and usually not written). a^{2} ≥ 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} = a^{n}/b^{n}.

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} = 5^{3}/4^{3} = 125/64
$8000 × 125/64 = 1000 × 125/8 = 1000 × 15.625 = 15625
The value of the investment in 3 years will be 15625.