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# Division with Exponents

by Ron Kurtus (revised 14 November 2015)

When you * divide two exponential expressions* with the same

**base**, you subtract the

**exponents**.

Note: Thebaseof the exponential expressionxis^{y}xand theexponentisy.

When you divide an exponential expression by itself, the exponent is **0**. When you divided by a larger exponential expression with the same base, the exponent is negative.

Questions you may have include:

- How do you divide numbers raised to a power?
- What does a 0 exponent designate?
- What does a negative exponent designate?

This lesson will answer those questions.

## Subtract exponents in division

When you divide exponential numbers or variables with the same base, you *subtract* the exponents.

### Numbers

This can be demonstrated with the example of dividing **7*7*7*7*7** by **7*7**.

The result is:

(7*7*7*7*7)/(7*7) =

7*7*7 = 7^{3}

Since **7*7*7*7*7** = **7 ^{5}** and

**7*7**=

**7**, you get

^{2}

7^{5}/7^{2}= 7^{5−2}= 7^{3}

### Variables

Likewise, if you divide **x ^{25}** by

**x**, you get

^{10}

x^{25}/x^{10}= x^{25-10}= x^{15}

If the variable is raised to some power of another variable, your still subtract the exponents:

x^{y}/x^{z}= x^{y−z}

### Must have same base

Note that the base must be the same or a multiple in order to reduce the expression by subtracting exponents.

Neither **5 ^{5} ÷ 2^{3}** nor

**x**can be reduced by this method, since the base of each is not the same.

^{7}/y^{4}However, in some cases, you can be clever and reduce the numbers to a common base.

Consider **6 ^{5} ÷ 2^{3}**. Since

**6**=

^{5}**(2*3)**=

^{5}**2**, then

^{5}*3^{5}

6=^{5}÷ 2^{3}2=^{5}*3^{5}/2^{3}2=^{5 − 3}*3^{5}2^{2}*3^{5}

## Dividing by itself

What happens when you divide an exponential number by itself?

11^{3}/11^{3}= 11^{3−3}= 11^{0}

The number **11 ^{0}** looks strange, but realizing that a number divided by itself equals

**1**, you can see that

**11**.

^{0}= 1### Case of x^{0}

**Rule**: Any number raised to the **0** power equals **1**.

Thus **x ^{0} = 7^{0} = 250^{0} = 1**.

### Case of 0^{0}

But what about **0 ^{0}**?

That is a very special case. Although it does not seem logical, most definitions say that **0 ^{0} = 1**.

The way to look at it is by examining fractions to the **0** power.

1/2^{0}= 1/2000^{0}= 1/2000000^{0}= 1/1 = 1

Thus, as the fraction gets smaller and smaller—approaching zero—its value remains at **1**.

## Negative exponents

But what happens when you divide by a number that is larger? If you divide **5 ^{3}** by

**5**, you will get

^{7}**5**.

^{3−7}= 5^{−4} But also, **5*5*5/5*5*5*5*5 = 1/5*5*5*5 = 1/5 ^{4}**. Thus,

**5**.

^{−4}= 1/5^{4}Likewise, **x ^{−3} = 1/x^{3}**.

Rule: A negative exponential is the reciprocal of the exponential.x.^{−y}= 1/x^{y}

## Summary

You subtract the exponents when dividing two exponential numbers or variables with the same base.

When you divide an exponential number by itself, the exponent is **0**.

When you divide by a larger exponential with the same base, the exponent is negative.

An exponential number with a negative exponent is the reciprocal of the exponential number.

Increase your understanding by knowing the rules

## Resources and references

### Websites

**Exponents: Basic Rules** - PurpleMath.com

**Exponent Rules** - RapidTables.com

**Exponents Calculator** - CalculatorSoup.com

### Books

## Questions and comments

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exponents_division.htm**

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## Where are you now?

## Division with Exponents