The exponent calculator will calculate the value of any base raised to any power. This page will cover all the related topics, including the negative exponent. Let's start with the basics.
What is an exponent?
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.
It's easy with small numbers, but for bases that are large numbers, decimals, or when they are raised to a power that's very large or negative, use our tool. If you wish to do exponentiation by hand, do the following:
- Determine the base and the power it's raised to, for example
- Write the base the same number of times as the exponent.
3 3 3 3 3
- Place a multiplication symbol between each base.
3 * 3 * 3 * 3 * 3.
3 * 3 * 3 * 3 * 3 = 243.
Negative exponent calculator
The concept is rather simple when the exponent is positive, but what happens when the exponent is negative? By the definition, if it is -2, we would multiply the base times itself negative two times. In actuality, what is happening here, we take the reciprocal of the base and change the negative exponent to positive and proceed as usual. If you'd like to work it out by hand, do the following:
- Determine the base and the exponent.
- Write the reciprocal of the base and change the sign of the exponent to positive
- Write the reciprocal of the base the same number of times as the exponent.
- Place a multiplication symbol between each.
- Multiply and get the result.
Here's a quick example:
5⁻⁴ = (1/5)⁴ = (1/5) * (1/5) * (1/5) * (1/5) = 1/625 = 0.0016
Squaring a base (raising a number to the power of 2) and taking the square root are similar concepts, many people consider one the opposite or 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.
Any number raised to the power of 0 equals 1. The negative exponent calculator is useful when dealing with exponential decay, which has a negative exponent in its formula.
What is 6 with an exponent of 4?
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.
How can I multiply exponents?
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:
- Add 3 + 5 = 8.
- Write the result as 28.
- Calculate it as 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256.
How can I divide exponents?
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 37-4 = 33 = 3 × 3 × 3 = 27.
How can I do fractional exponents?
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, 21/2 is equal to √2, 21/3 is ³√2, 21/4 is ∜2, and so on.
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