This Exponent Calculator is an online tool for calculating the value of any base number raised to power. In the following text you will learn more about exponents, negative exponents, basic properties and other interesting things.

If you want to learn more about logarithms, head to our Log Calculator with steps. Math is really interesting, and here you can learn more about various subjects, such as Cross Product and Trapezoid, or maybe Cofunction, Distance Calculator, and Linear Equation as well.

What is an exponent?

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. It is said the base is raised to the power of the exponent.

Exponent and exponentiation
3 raised to the power of 2

With our calculator, if you want to calculate the exponent, all you have to do is enter the base and the result. If you want to calculate the base, you need to enter the exponent and the result. And of course, if you want to calculate the power, all you have to do is enter the base and the exponent.

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}}

Negative exponents

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

Properties (rules) of exponents

Provided the base of a power is not 0, there are certain rules that you can use to make calculations easier.

If two or more powers have the same base, you have very simple methods when it comes to multiplying exponents and dividing exponents. 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}

There is also a rule for exponentiating an exponent. Essentially, if a number has an exponent, and the exponent has an exponent, those two exponents can be multiplied in order to simplify the calculation:

 (b^m)^n = b^{m \cdot n} 

Adding exponents

In order for you to be able to add up powers, both the bases and the exponents need to be the same. The only thing that is going to change is the coefficient. The coefficient is the number that stands before the base, and serves as a multiplier for the power. So, a power with a coefficient would look like this:

4x^3

Now, in any case where both the bases and the exponents are the same, you can just add up the coefficients:

 3x^5 + 6x^5 + 7x^5 = 16x^2 

As we can see, neither the base nor the exponent changed. The only part that changed was the coefficient.

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. Also, there is one more specific exponent, it’s the exponential value of the Euler’s number e. To learn more about this, head to our e Calculator – eˣ.

Powers of 10

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

10^{4}=10000

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

So, 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 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!

Other Calculators

Feel free to visit other sections and categories of our website as well. People are often interested in finances, and want to calculate markdown value with our Markdown Calculator, or GDP per Capita values, for example. Beside this Exponent Calculator, and other mentioned ones, you can see our popular Hooke’s Law Calculator, and Relative Standard Deviation – RSD Calculator.