Skill: Apply the quotient rule for exponents

Explanation and Free Practice Resources

For nonzero bases, the quotient of powers with the same base is simplified by subtracting the exponent in the denominator from the exponent in the numerator: $a^m/a^n=a^{m-n}$ . This reasoning connects division with inverse powers and supports simplifying algebraic expressions and scientific notation; the exponents are not divided, and more advanced extensions to arbitrary real or complex exponents are not included.

On this page:

Learn “Apply the quotient rule for exponents” with a detailed explanation of the concept
Master “Apply the quotient rule for exponents” with unlimited, free practice problems
Download (or modify) a free worksheet on Apply the quotient rule for exponents you can use at home or in class with practice problems that scaffold and explore “Apply the quotient rule for exponents” from different perspectives.
Start working on adaptive units like “Exponents - Division - Intro” or “Exponents - Multiplication and Division - Practice” that introduce, teach, or practice “Apply the quotient rule for exponents” to maximize your learning.

Detailed Explanation: Apply the quotient rule for exponents

When dividing powers with the same nonzero base, keep the base and subtract the denominator’s exponent from the numerator’s exponent:

\frac{a^m}{a^n}=a^{m-n}

The exponents are subtracted, not divided.

Example:

\frac{5^7}{5^3}

The bases are the same: both are $5$ .
Subtract the denominator’s exponent from the numerator’s exponent:

7-3=4

Keep the base $5$ :

\frac{5^7}{5^3}=5^{7-3}=5^4

If needed, evaluate:

5^4=625

So,

\boxed{\frac{5^7}{5^3}=5^4=625}

Learn by doing: Apply the quotient rule for exponents

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Your results

Practice with unlimited practice problems

Exponents - Division - Positive by Positive to Positive

Earned ?

Download/Modify Free Grade 10 'Apply the quotient rule for exponents' Worksheet

Print this free 'Apply the quotient rule for exponents' worksheet to practice this skill away from the computer. Click 'Modify' to easily customize this worksheet to your exact needs.

Units Teaching This Skill

These adaptive units introduce, teach, or reinforce this skill (Apply the quotient rule for exponents) through scaffolded problems that grow in difficulty as concepts are understood.