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.

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

base^{power} or base^{exponent}

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, 3^{4}. 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.

0^{n} = 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.

0^{0} = 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.

x^{a} . x^{b} = x^{(a+b)}

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

x^{a} / x^{b} = x^{(a-b)}

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

x^{-a} = 1/(x^{a})

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