Exponents and Square Roots

Exponentiation is a mathematics word that means applying an exponent to a base or raising a number to a power. The square root of a number is another number that when multiplied by itself yields the original number.

Base and Power

An expression in exponentiation has two parts, a base and a power. The power is often called the exponent.

basepower or baseexponent

The base is a number you raise to a power. The power is a number you raise the base by (it is indicated by a raised number - a superscript). For example, 34. Here, 3 is the base and 4 is the exponent. You say this number as "three raised to the fourth power" or "three to the fourth".

When you raise a number to the second power (power of 2), you call it square measure. When you raise a number to the third power (the power of 3) you call it cubic  measure.

When you work with exponentiation, you can use almost any base and almost any power. However, you need to know about a few special bases and powers.

Powers with base 10

Using powers of 10 is a much easier way to do math on very large or very small numbers. In exponentiation, the equivalent of a decimal shift is increasing or decreasing the power. A one-decimal shift to the right gets you a tenfold increase in the value of a number. That is, to multiply a number with a base of 10 by 10, just increase its exponent by 1. A one-decimal shift to the left produces a tenfold decrease in the value of a number. To divide a number with a base of 10 by 10, decrease its exponent by 1. These shifts mean you are changing the order of magnitude.

Powers with base 2

If you work with computers, various base 2 numbers are your daily companions. For example, Processor speed in Hz, RAM in gigabytes, Disk capacity, and so on.

Powers with base 1

What happens when you elevate 1 to various powers? 1 to any power equals 1.

Powers with base 0

If the exponent is positive, the power of zero is zero.

0n = 0, no matter how large the exponent.

If the exponent is negative, the power of zero is called "undefined", because division by zero is implied and that is impossible.

0-n = undefined

If the exponent is zero, some mathematicians define it as one and others leave it undefined.

00 = 1 or may be undefined

Powers with base -1

When the exponent n even, (-1)n = 1

When the exponent n is odd, (-1)n = -1

Powers of 1 and 0

Any number raised to the power of 1 is itself.

Any number raised to the power of 0 is 1.


1. When you multiply two terms with exponents, the result is identical to adding the exponents.

xa . xb = x(a+b)

2. The same idea is true when you divide. You just subtract the powers.

xa / xb = x(a-b)

3. If the result turns out to be a negative exponent, it is a reciprocal. 

x-a = 1/(xa)

Square Roots

The inverse of squaring a number is finding a number's square root. A square root is all about finding the value of the base when you know only the result of squaring the value.

For example, the square root of 16 is 4.

√16 = 4