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# Division with Exponents

by Ron Kurtus (revised 17 August 2012)

When you divide two exponential numbers or variables with the same base, you subtract the exponents. When you divide an exponential number by itself, the exponent is **0**. When you divided by a larger exponential 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 smaler and smaller—apporaching 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

### Books

## Questions and comments

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## Where are you now?

## Division with Exponents