Exponent Rules Calculator (Product, Quotient, Power)

An exponent rules calculator simplifies powers by applying the laws of exponents one step at a time. With the same base, multiply by adding exponents, divide by subtracting them, and raise a power to a power by multiplying them: 2³ × 2⁴ = 2⁷ = 128, and (2x³y)² × x⁻⁴ = 4x²y².

Embed this calculator on your site

Free to use on class pages, blogs and course pages. Copy the code below and paste it into your page:

<iframe src="https://www.effortlessmath.com/math-topics/exponent-rules-calculator/?em_embed=1" width="100%" height="720" style="border:0;max-width:100%" title="Exponent Rules Calculator (Product, Quotient, Power) by Effortless Math" loading="lazy"></iframe><p><a href="https://www.effortlessmath.com/math-topics/exponent-rules-calculator/">Exponent Rules Calculator (Product, Quotient, Power)</a> by <a href="https://www.effortlessmath.com/">Effortless Math</a></p>Copy code

How to use the exponent rules calculator

1. Simplify mode: type an expression with numbers and letters, such as (3a^2b)^3/9a^4 or (16x^8y^-4)^(3/4). Use ^ for powers and put fractional exponents in parentheses.
2. Two powers mode: enter a base and two exponents, then choose multiply, divide or power. This shows the product, quotient or power rule on its own, with the numeric value.
3. One power mode: enter a base and a zero, negative or fractional exponent to see its exact value, such as 8^−2/3 = 1/4.

The answer is written with positive exponents, followed by the same answer with negative exponents and, when there are fractional exponents, in radical form. A number check substitutes small primes for the letters and confirms that the original and the answer agree. For plain number powers such as 3.7^2.5, the numeric exponent calculator is the better tool.

The exponent rules

Every law of exponents comes from one idea: bⁿ means n factors of b multiplied together. For a nonzero base b and rational exponents m and n:

• Product rule: b^m · bⁿ = b^{m + n}. Same base, multiply, add the exponents.
• Quotient rule: b^m ÷ bⁿ = b^{m − n}. Same base, divide, subtract the exponents.
• Power rule (power of a power): (b^m)ⁿ = b^mn. Multiply the exponents.
• Zero exponent rule: b⁰ = 1, because bⁿ ÷ bⁿ = b⁰ and anything nonzero divided by itself is 1.
• Negative exponent rule: b⁻ⁿ = 1/bⁿ. A negative exponent means a reciprocal, not a negative number.
• Fractional exponent rule: b^m/n = (ⁿ√b)^m. The denominator is the root and the numerator is the power.

All the laws of exponents with examples

Worked examples: using several exponent rules at once

Example 1: (2x³y)² × x⁻⁴

Power of a product rule first: (2x³y)² = 2² · x^3·2 · y² = 4x⁶y². Then the product rule for x: x⁶ · x⁻⁴ = x^{6 + (−4)} = x². The answer is 4x²y². A common error is forgetting to square the 2, which gives 2x²y².

Example 2: 6a⁵ ÷ 3a⁻²

Divide the numbers, 6 ÷ 3 = 2. Quotient rule for a: a^{5 − (−2)} = a⁷. The answer is 2a⁷. Subtracting a negative exponent adds it, which is where most sign slips happen.

Example 3: (16x⁸y⁻⁴)^3/4

Raise each factor to the 3/4 power. 16^3/4 = (⁴√16)³ = 2³ = 8, x^8·3/4 = x⁶ and y^−4·3/4 = y⁻³. With positive exponents the answer is 8x⁶/y³.

Study tip: apply one rule at a time and write the rule’s name next to each line. If two lines in a row change more than one thing, a step was skipped.

Common mistakes with exponent rules

Checking homework with the exponent rules calculator

Work the problem by hand first and name each rule as you use it. Then type the same expression and compare your lines with Show steps. If the answers differ, use the number check: substitute x = 2 (and y = 3) into your answer and into the original. If the two numbers do not match, your answer is wrong; if they match, the difference is only the form, for example x⁻² versus 1/x². Most teachers want the final answer with positive exponents only.

Exponent rules practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

What are the exponent rules?

The main rules are the product rule (add exponents), the quotient rule (subtract exponents), the power rule (multiply exponents), the power of a product and of a quotient, the zero exponent rule (b⁰ = 1), the negative exponent rule (b⁻ⁿ = 1/bⁿ) and the fractional exponent rule (b^m/n is the nth root of b^m).

What is the exponent power rule?

When a power is raised to another power, multiply the exponents: (b^m)ⁿ = b^mn. For example (x³)⁴ = x¹². Do not confuse it with the product rule, x³ · x⁴ = x⁷, where the exponents are added.

What is the negative exponent rule?

A negative exponent means take the reciprocal: b⁻ⁿ = 1/bⁿ. So 5⁻² = 1/25, and a factor with a negative exponent in the denominator moves to the numerator: 1/x⁻³ = x³. The result is not negative.

What is the zero exponent rule?

Any nonzero base to the power 0 equals 1: 7⁰ = 1 and (3xy)⁰ = 1. Watch the parentheses: 3x⁰ = 3 · 1 = 3, because only x has the 0 exponent. The expression 0⁰ is left undefined in algebra.

How do exponent rules work with fractions?

A power of a fraction applies to the top and bottom: (2/5)² = 4/25. A negative exponent flips the fraction: (2/5)⁻² = (5/2)² = 25/4. A fractional exponent is a root: 16^3/4 = (⁴√16)³ = 2³ = 8.

How do you simplify expressions using exponent rules?

Remove parentheses with the power rules, combine the same bases with the product and quotient rules, then rewrite negative exponents as fractions. For (3a²b)³ ÷ 9a⁴: (3a²b)³ = 27a⁶b³, then 27 ÷ 9 = 3 and a^{6 − 4} = a², giving 3a²b³.

Related calculators

• Exponent calculator: numeric powers such as 3.7^2.5
• Scientific notation converter: powers of ten in practice
• Radical simplifier: simplify roots that come from fractional exponents
• Logarithm calculator: solve for an unknown exponent
• Order of operations calculator: where exponents fit in PEMDAS