# Multiplying Exponents Calculator

By Anna Szczepanek, PhD
Last updated: Apr 20, 2021

Come and use Omni's multiplying exponents calculator whenever you need help in problems related to exponentiation. This calculator can show you step-by-step how to multiply exponents using the most important rules, such as the product of power rule and the power of product rule.

## What is an exponent?

Exponentiation is an important mathematical operation. Exponents appear in many different fields, ranging from quadratic equations (and polynomials in general) to probability.

Recall that an exponent is a way of expressing repeated multiplication. For example, `74` means that we want `7` multiplied by itself `4` times:

`74 = 7 * 7 * 7 * 7 = 2401`

The number that gets multiplied (`7` in the example above) is called the base. The number that tells how many times the base is multiplied by itself (`4` in the example above) is called the exponent.

## Non-positive exponents

We've said above that the exponent tells us how many times to multiply the base by itself. However, there are special cases that need further explanation; namely, what to do if the exponent is equal to zero or some negative number?

Zero exponent. If `a ≠ 0`, then

`a0 = 1`

In words: Any non-zero base raised to the power of zero is equal to one.

 ❌ Zero to the power of zero `00` is an indeterminate symbol. In other words, it doesn't make sense to perform this calculation.

Negative exponent. If `a ≠ 0`, then

`a-n = 1 / an`

In words: To determine a negative exponent, ignore the minus sign for a while and raise the base to the positive power. Then take the reciprocal.

We can also say that negative exponents represent repeated division by the base, that is, repeated multiplication by the reciprocal of the base:

`a-n = 1 / (a * ... * a) = (1/a) * ... * (1/a)`

## How to use Omni's multiplying exponents calculator?

Here is a short instruction explaining the use of our tool:

1. Enter the bases `x` and `y` and the exponents `a` and `b`.

2. Our multiplying exponents calculator shows step-by-step how to determine the product `xa * yb`

3. The calculator also shows how to multiply negative exponents. In such case, there might be fractions in the final result, and you might want to find their decimal approximations. To turn this option on, go to the `advanced mode`.

4. In the `advanced mode` you can also adjust the precision with which the approximations are displayed. By default, our calculator uses 5 significant figures.

## How to multiply exponents? Multiplying exponents rules

Well, the best way to multiply exponents is to use our multiplying exponents calculator! 😉 In case you ever needed to perform the multiplication of exponents by hand, here we list the most useful rules:

• Product of powers rule:

`wa * wb = wa + b`

In words: add the exponents if the bases are the same.

• Power of a product rule:

`wa * ua = (w * u)a`

In words: multiply the bases if the exponents are the same.

Going in the other direction, i.e., from right to left, we can express the rule as follows: if the base is written as a product of several factors, then we can distribute the exponent to each of these factors.

• Power of powers rule:

`(wa)b = wa * b`

In words: when raising an exponent to another exponent, multiply the two exponents.

In the remaining part of the article, we show you a few examples of how to multiply exponents with the help of these rules.

## Multiplying exponents with the same base

Do the exponents you want to multiply have the same bases? Lucky you! This is the easiest case of multiplying exponents! The only thing you have to remember is to add the exponents together. In the following example we will show you how to do it.

Example 1. Let's solve `711 * 7-9`

1. We see that the bases are both equal to `7`, and the exponents are `11` and `-9`. We use the product of powers rule and sum the two exponents:

`711 * 7-9 = 711 + (-9) = 72`

2. So we obtain:

`711 * 7-9 = 7 * 7 = 49`

## Multiplying exponents with different bases

If the bases are different, then your task is a tiny bit more complicated, but you can manage! Let's discuss two examples from which you can learn how to multiply exponents with different bases.

Example 2. Let's compute `54 * 24`

1. As we can see, the exponents are the same and equal to `4`, while the bases are `5` and `2`. This means we can use the power of a product rule, that is, we can multiply the bases together:

`54 * 24 = (5 * 2)4 = 104`

2. Recall that raising `10` to any power is very simple: after the leading `1` we write down as many zeros as the exponent says. In this case, it's four zeros:

`54 * 24 = 104 = 10000`

What should you do when the exponents are different and the bases are also different? Unlike the two special cases we've discussed so far, there’s no quick way.

The best you can do is to simplify the problem by using the three rules given above along with the prime factorization of both bases. Prime factorization means that we represent a number as a product of prime numbers raised to suitable powers. For instance, the prime factorization of `24` is the following:

`24 = 2 * 2 * 2 * 3 = 23 * 3`

Example 3. Let's discuss how to deal with `24-4 * 903`.

1. The bases are `24` and `90` and exponents are `-4` and `3`. Let's perform prime factorization on the bases:

`24 = 2 * 2 * 2 * 3 = 23 * 3`

`90 = 2 * 3 * 3 * 5 = 2 * 32 * 5`

2. We apply the power of a product rule:

`24-4 = (23 * 3)-4 = (23)-4 * 3-4`

`903 = (2 * 32 * 5)3 = 23 * (32)3 * 53`

3. We apply the power of powers rule:

` (23)-4 = 23 * (-4) = 2-12 `

`(32)3 = 32 * 3 = 36`

4. As a result, we obtain:

`24-4 = 2-12 * 3-4`

`903 = 23 * 36 * 53`

Hence:

`24-4 * 903 = 2-12 * 3-4 * 23 * 36 * 53`

5. We apply the product of powers rule:

`2-12 * 23 = 2-12 + 3 = 2-9`

`3-4 * 36 = 3-4 + 6 = 32`

It follows that:

`24-4 * 903 = 2-9 * 32 * 53`

6. This is the simplest form we can get. If you want to get rid of the exponents, calculate the factors one by one:

`2-9 = 1 / 29 = 1 / 512`

`32 = 9`

`53 = 125`

Finally, we obtain:

`24-4 * 903 = (1 / 512) * 9 * 125 = 1125/512 ≈ 2.1973`

Anna Szczepanek, PhD
This calculator simplifies and computes the product of exponents:
xa ⋅ yb
Base x
Exponent a
Base y
Exponent b
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