Exponent Rules

(updated 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.

This lesson will answer those questions.

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Exponent product rule

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

x⁴ * x² = x^{4 + 2} = x⁶

2³ * 2⁵ = 2^{3 + 5} = 2⁸

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

When you multiply two exponential expressions with the same exponent, the product of the bases is raised to that exponential power:

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

3³ * 2³ = (3 * 2)³ = 6³

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:

x⁵/x² = x^{5 − 2} = x³

2⁷/2⁵ = 2^{7 − 5} = 2²

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:

y³/y⁵ = y^{3 − 5} = y⁻² = 1/y²

3⁶/3⁸ = 3^{6 − 8} = 3⁻² = 1/3² = 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

3³/2³ = (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

(7²)³ = 7⁶

Proof:

(7²)³ = (7²)(7²)(7²) = 7^{2 + 2 + 2} = 7⁶

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Learn the rules and follow them

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Resources and references

Websites

Exponents: Basic Rules – PurpleMath.com

Exponent Rules – RapidTables.com

Laws of Exponents – MathisFun.com

Exponents Calculator – CalculatorSoup.com

Algebra Resources