Exponentiation is a mathematical function that has two numbers: b, which is the base, and n which is the exponent, and the result is the power.

Base, exponent and power

In cases where the exponent is a positive number (0<), it will represent the number of times the base is multiplied by itself:

b^{n} = \underbrace{b\times ... \times b}_{\text{$n$ times}}

When the exponent is a negative number (<0), you essentially calculate the reciprocal value of the base:

b^{-n} = \frac {1} {b^{n}}

If the base is a negative number, whether the power will be a positive or a negative number depends on the exponent. In order for the power to be a positive number, the exponent has to be an even number, because of the rules when it comes to multiplying negative numbers.

-5^{4}=-5\cdot (-5)\cdot (-5)\cdot (-5)=625

Contrary to this, if the exponent is an odd number, the power will be negative:

-5^{3}=-5\cdot(-5)\cdot (-5)=-125

Multiplying and dividing exponents

If two or more powers have the same base, you have very simple methods when it comes to multiplying and dividing them. When you’re multiplying two powers with the same base, you can just copy the base and add up the exponents:

b^{m} * b^{n} = b^{m+n}

When it comes to the division of powers that have the same base, you once again just copy the base, but this time, you subtract the exponents instead of adding them up.

\frac {b^{m}} {b^{n}} = b^{m-n}

Specific exponents

It should be obvious that any number exponentiated to the power of 1 stays the same. Something that is perhaps less known is that any number exponentiated to the power of 0 is equal to 1. We can easily prove this.

Let’s say we have two powers with the same base, and the same exponent. If we divide these two powers, the base will stay the same and the exponent will be zero. However, we also know that any number divided by itself is always equal to 1.

\frac {15^{5}} {15^{5}}=15^{0}=1

This proves that any number to the power of 0 is equal to 1. However, people are still debating whether 0 to the power of 0 is 1 or simply undefined.

If a power has 2 as its exponent, that number is squared, and if it has 3 as its exponent, it is cubed.

Powers of 10

10 to the power of any positive integer (n) can 1 with n zeroes:


If the number 10 has a negative exponent, we can write it as a decimal number that has n-1 zeroes after the period, with a 1 at the nth place:

10^{-4}= 0.0001

We often use the powers of 10 in scientific notations to describe very small or large values. For example, the weight of an electron is as 9.11×10^31 kg. As well as that, the SI system uses them for a variety of prefixes.
For more info, be sure to check out our Scientific Notation Calculator!