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# Exponent Rules

by Ron Kurtus (revised 18 January 2022)

When you perform algebraic operations on exponential expressions, there are several * exponent rules* that you can follow. These rules are the product rule, quotient rule, and power rule.

Questions you may have include:

- What is the product rule?
- What is the quotient rule?
- What is the power rule?

This lesson will answer those questions.

## Exponent product rule

When you multiply two exponential expressions with the **same base**, you simply add the exponents:

When you multiply two exponential expressions with the

x^{4}* x^{2}= x^{4 + 2}= x^{6}

2^{3}* 2^{5}= 2^{3 + 5}= 2^{8}

y^{a}* y^{b}= y^{a + b}

**same exponent**, the product of the bases is raised to that exponential power:

x^{a}* y^{a}= (xy)^{a}

3^{3}* 2^{3}= (3 * 2)^{3}= 6^{3}

## Exponent quotient rule

When you divide two exponential expressions with the **same base**, you simply subtract the denominator (lower) exponent from the numerator (upper) exponent:

x^{5}/x^{2}= x^{5 − 2}= x^{3}

2^{7}/2^{5}= 2^{7 − 5}= 2^{2}

y^{a}/y^{b}= y^{a − b}

If the denominator (lower) exponent is **greater** than the numerator (upper) exponent, the result will have a negative exponent:

y^{3}/y^{5}= y^{3 − 5}= y^{−2}= 1/y^{2}

3^{6}/3^{8}= 3^{6 − 8}= 3^{−2}= 1/3^{2}= 1/9

When you divide two exponential expressions with the **same exponent**, the quotient of the bases is raised to that exponential power:

x^{a}/ y^{a}= (x/y)^{a}

3^{3}/2^{3}= (3/2)^{3}

## Exponent power rule

When you raise an exponential expression to a power, you simply multiply the exponents:

(y^{a})^{b}= y^{ab}

(7^{2})^{3}= 7^{6}

Proof:

(7^{2})^{3}= (7^{2})(7^{2})(7^{2}) = 7^{2 + 2 + 2}= 7^{6}

## Summary

When you perform algebraic operations on exponential expressions, there are several exponent rules that you can follow. These rules are the product rule where you add the exponents, the quotient rule where you subtract the denominator exponent from the numerator exponent, and the power rule where you multiply the exponents.

Learn the rules and follow them

## Resources and references

### Websites

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

**Exponent Rules** - RapidTables.com

**Laws of Exponents** - MathisFun.com

**Exponents Calculator** - CalculatorSoup.com

### Books

(Notice: The *School for Champions* may earn commissions from book purchases)

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## Exponent Rules