Exponent Calculator

The exponent calculator calculates the value of any base raised to any power. This page covers all the related topics, including laws of exponents and the negative exponent. Let's start with the basics.

An exponent is a way to represent how many times a number, known as the base, is multiplied by itself. It is represented as a small number in the upper right hand corner of the base. For example: x² means you multiply x by itself two times, which is x × x. Likewise, 4² = 4 × 4, etc. If the exponent is 3, in the example 5³, then the result is 5 × 5 × 5.

If you wish to do exponentiation by hand, do the following:

1. Determine the base and the power it's raised to, for example, 3⁵.
2. Write the base the same number of times as the exponent. 3 3 3 3 3
3. Place a multiplication symbol between each base. 3 × 3 × 3 × 3 × 3.
4. Multiply! 3 × 3 × 3 × 3 × 3 = 243.

Taking powers is easy with small numbers, but it gets tricky if the base is large or a decimal, or if the exponent is large, negative, or fractional. Don't hesitate to use our tool in such cases!

What happens if we want to multiply two powers with the same base together? We simply add the exponents! Consider, for instance, the product 5³ × 5². This is simply

5³ × 5² = (5 × 5 × 5) × (5 × 5) = (5 × 5 × 5 × 5 × 5) = 5⁵

Since we first multiply 3 copies of 5 and then 2 copies of 5, in total, we multiply 5 copies of 5. This property extends more generally, giving us the first law of exponents:

xⁿ × x^m = x^{n + m}

We divide powers similarly: this time, we subtract the exponents. Consider, for instance, 5³/5². This is:

5³/5² = (5 × 5 × 5)/(5 × 5) = 5 = 5¹

since two copies of 5 from the divisor cancel with two copies of 5 from the dividend, leaving only one 5 remaining. This gives us the second law of exponents:

xⁿ/x^m = x^{n - m}

It is all relatively simple when the exponent is positive, but what if the exponent is zero, 0, or negative? It turns out that we can consider such exponents as long as they conform to the two laws of exponents from the previous section!

Let us look first at the zero exponent: we define 5⁰ = 1, the same with 5 replaced by any positive number x. The logic behind this is that we want it to satisfy the first law of exponents, i.e., we want the identity 5ⁿ × 5⁰ = 5^{n + 0} = 5ⁿ to hold. This is only possible if 5⁰ = 1!

What if the exponent is negative, e.g., 5^-4? Then, we swap the base 5 with its reciprocal 1/5 and change the negative exponent to positive so that 5^-4 = (1/5)⁴. From then on, we proceed as usual. If you'd like to work it out by hand, here is a complete recipe:

1. Determine the base and the exponent.
2. Write the reciprocal of the base and change the sign of the exponent to positive.
3. Write the reciprocal of the base the same number of times as the exponent.
4. Place a multiplication symbol between each.
5. Multiply and get the result.

In our example, we get:

5^-4 = (1/5)⁴ = (1/5) × (1/5) × (1/5) × (1/5) = 1/625 = 0.0016

Why are powers with negative exponents defined this way? By the second law of exponents, we want:

5⁴ × 5^-4 = 5^{4 - 4} = 5⁰ = 1

and this is only possible if 5^-4 = 1/5⁴ = (1/5)⁴.

Give negative exponents a try in the exponent calculator.

Want to see an example of how exponents show up in real life? Welcome to the world of bacteria! Bacteria proliferate by cell doubling, meaning that a bacterium cell splits into two child cells. Different bacteria species split at different time intervals: for instance, under optimal conditions, the E. coli bacteria double every 20 minutes.

That means that within an hour, each E. coli bacterium will split into 8 child cells: there will be 2 of them following the first cell doubling after 20 minutes, 4 cells after 40 minutes, and 8 of them after an hour. This can be computed using exponents:

• An hour can be broken down into 3 intervals of 20 minutes;
• After each 20-minute period, every cell divides into two; hence
• The number of cells after an hour is 2³ = 8.

What if we leave our single E. coli bacterium cell for 10 hours and let it divide unhindered? The period of 10 hours consists of 30 intervals of 20 minutes each. Hence, by the end of the 10-hour period, our cell will have divided into 2³⁰ = 1,073,741,824 (over one billion) cells!

Squaring a base (raising a number to the power of 2) and taking the square root are similar concepts; in fact, they are opposite operations, which means one is the undoing of the other. If you want to square the number 6, you take 6 × 6 = 36. Now, if you want to find what two identical numbers multiply to give you 36, you take the square root of 36. This square root gives the value of 6. It can also be noted that squaring a square root removes the radical.

Likewise, cubing a base (raising a number to the power of 3) will give us a perfect cube. In case you need to calculate the cube root, you can use our cube root calculator, which is an excellent tool that will calculate the cube root of any number.

In modular arithmetic, there are dedicated methods of exponentiation – learn more with the power mod calculator.

Besides, you may check our logarithm calculator, which is the inverse function of the exponent.

1296. To calculate 6 with an exponent of 4, write it as 6⁴ and multiply four instances of 6 together. It can be written as 6 × 6 × 6 × 6 = 1296.

If you want to multiply the exponents, make sure they have the same base. Then, simply add the original exponents to find the new exponent of the product. For example, to multiply 2³ by 2⁵:

1. Add 3 + 5 = 8.
2. Write the result as 2⁸.
3. Calculate it as 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256.

You can also divide exponents with the same base, subtracting the exponents. To check it out, let's divide 3⁷ by 3⁴. Subtract the exponents to obtain 3^7-4 = 3³ = 3 × 3 × 3 = 27.

A fractional exponent is one in which the exponent of a number is a fraction. The general rule is that a fractional exponent like 1/n means to take the n-th root of a number. For example, 2^1/2 is equal to √2, 2^1/3 is ³√2, 2^1/4 is ∜2, and so on.