# Dividing Exponents Calculator

Created by Anna Szczepanek, PhD
Reviewed by Dominik Czernia, PhD and Jack Bowater
Last updated: Sep 30, 2022
Table of contents:

Omni's dividing exponents calculator is here to serve you whenever you need to compute a quotient of two exponents. With our calculator, you can learn step-by-step how to divide exponents. Do you need help dividing exponents with the same base or rather exponents with different bases? Or maybe dividing negative exponents? Scroll down to learn more and see a few examples!

Also, don't forget to check out Omni's multiplying exponents calculator!

## What is an exponent?

Recall form our exponentiation calculator that exponents are a shortcut to express repeated multiplication. For example, 58 means that 5 is multiplied by itself 8 times:

$5^8 = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5$
The number that we multiply is a base and the number that tells how many times we multiply the base by itself is called an exponent.
In the example above, 5 is the base and 8 is the exponent.

#### Non-positive exponents

What we've said above about repeated multiplication makes sense only for positive integer exponents. For the case of an exponent equal to zero or to a negative integer, we need extra definitions.

• Zero exponent. We have $a^0 = 1$ for any $a \neq 0$.

⚠️ Mind the assumption! We cannot raise zero to the power of zero!

• Negative exponent. For a negative integer $n$ we define $a^n = 1 / a^{-n}$, where we assume that $a \neq 0$ (because we don't want to divide by zero). That is, a number raised to a negative exponent is the reciprocal of the number raised to the opposite exponent (in other words, to the absolute value of the original exponent).

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

Here is a summary of the key features of our dividing exponents calculator:

1. The calculator determines the quotient $x^a / y^b$. You need to input data, i.e., the bases $x$ and $y$ and the exponents $a$ and $b$ into the respective fields.

2. Our dividing exponents calculator shows a step-by-step solution to your problem.

3. If there are fractions in the final result, our calculator can find their decimal approximations. To enable this option, go to the advanced mode.

4. In the advanced mode, you can also increase the precision with which the dividing exponents calculator approximates the fractions. By default, our calculator shows 5 significant figures.

## How to divide exponents? Dividing exponents rules

The most important rule if you want to determine the quotient of exponents by hand is the following quotient of powers rule:

$w^{a} / w^{b} = w^{a - b}$

The above formula is crucial when you want to divide exponents with the same base. Let's see how it works in practice:

Example 1. Let's compute $11^{9} / 11^{5}$.
We use the quotient of powers rule:

$11^{9} / 11^{5} = 11^{9 - 5} = 11^{4} = 14641$

Example 2. Let's compute $5^{2} / 5^{-3}$.
We again use the quotient of powers rule:

$5^{2} / 5^{-3} = 5^{2 - (-3)} = 5^{5} = 3125$

Example 3. Let's compute $7^{-8} / 7^{-6}$.
Are you surprised that we use the quotient of powers rule?

$7^{-8} / 7^{-6} = 7^{-8 - (-6)} = 7^{-2} = 1 / 7^{2} = 1/49$

As you can see, it's really simple to divide exponents with the same bases! The only thing you have to remember is to subtract the exponents.

In the next section, we show you several examples of how to divide exponents with different bases.

## Dividing exponents with different bases

If the bases are different, then it is slightly more complicated to divide an exponent by another exponent. The strategy we're going to adopt is prime factorization of bases, which means that we will rewrite each basis as a product of prime numbers raised to suitable powers. Go to the prime factorization calculator to learn more.

We will also use the power of powers rule:

$(w^{a})^{b} = w^{ab}$

Let's go together through some examples to see how to divide exponents with different bases.

Example 4. Compute $15^{4} / 12^{6}$

1. The bases are 15 and 12. Let's write down their prime factorization:

$15 = 3 \cdot 5$

$12 = 2 \cdot 2 \cdot 3 = 2^{2} \cdot 3$

2. We apply the power of a product rule:

$15^{4} = (3 \cdot 5)^{4} = 3^{4} \cdot 5^{4}$

$12^{6} = (2^{2} \cdot 3)^{6} = (2^{2})^{6} \cdot 3^{6}$

3. We apply the power of powers rule:

$(2^{2})^{6} = 2^{2 \cdot 6} = 2^{12}$

4. As a result, we obtain exponent divided by exponent:

$15^{4} / 12^{6} = (3^{4} \cdot 5^{4}) / (2^{12} \cdot 3^{6})$

5. We apply the quotient of powers rule:

$3^{4} / 3^{6} = 3^{4 - 6} = 3^{-2}$

It follows that:

$15^{4} / 12^{6} = 5^{4} \cdot 3^{-2} / 2^{12}$

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

$5^{4} = 625$

$3^{-2} = 1 / 3^{2} = 1/9$

$2^{12} = 4096$

Finally, we obtain:

$15^{4} / 12^{6} = 625 \cdot (1/9) \cdot (1/4096) = 625/36864 ≈ 0.01695$

Example 5. Compute $27^{-2} / 3^{-7}$

In this example, we face the challenge of dividing negative exponents. As you'll see, it's not hard at all!

1. The bases are 27 and 3. Obviously, 3 is prime, and the prime factorization of 27 reads:

$27 = 3 \cdot 3 \cdot 3 = 3^{3}$

2. We apply the power of powers rule:

$(3^{3})^{-2} = 3^{3 \cdot (-2)} = 3^{-6}$

3. As a result, we obtain:

$27^{-2} / 3^{-7} = 3^{-6} / 3^{-7}$

4. We apply the quotient of powers rule:

$3^{-6} / 3^{-7} = 3^{-6 - (-7)} = 3^{1} = 3$

5. That's it!

$27^{-2} / 3^{-7} = 3$

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