Quotient Rule for Exponents

Learn why dividing powers with the same base means subtracting the exponents, and use the quotient rule with numbers, variables and coefficients.

Intermediate20 min18 problems
Practice Problems

Definition

The quotient rule for exponents says: when you divide two powers with the same base, keep the base and subtract the exponents (exponent on top minus exponent on the bottom):
aman=am−n(a≠0)\frac{a^m}{a^n} = a^{m-n} \quad (a \neq 0)
Why it works: write both powers as repeated factors and cancel the common factors, because any number divided by itself is 11:
5652=5⋅5⋅5⋅5⋅5⋅55⋅5=54\frac{5^6}{5^2} = \frac{5 \cdot 5 \cdot 5 \cdot 5 \cdot \cancel{5} \cdot \cancel{5}}{\cancel{5} \cdot \cancel{5}} = 5^4
The 22 factors in the denominator cancel 22 of the 66 factors on top, so 6−2=46 - 2 = 4 factors stay.
Check with numbers: 56÷52=15625÷25=625=545^6 \div 5^2 = 15625 \div 25 = 625 = 5^4.
In this lesson the top exponent is always the larger one (m>nm > n). When both exponents are equal, every factor cancels (lesson Zero Exponent); when the bottom exponent is larger, factors stay in the denominator (lesson Negative Exponents).

Try it now

Simplify 7973\frac{7^9}{7^3}.

Worked Examples

Simplify 5652\frac{5^6}{5^2} and evaluate it.
1

Check the bases

Both powers have the base 55 → Same base, so the quotient rule applies
2

Subtract the exponents (top minus bottom)

6−2=46 - 2 = 4 → 545^4
3

Evaluate the power

5⋅5⋅5⋅55 \cdot 5 \cdot 5 \cdot 5 → 625625
4

Check by calculating both powers

56÷52=15625÷255^6 \div 5^2 = 15625 \div 25 → 625625, the same result

Common Mistakes

Dividing the exponents: x8x2=x4\frac{x^8}{x^2} = x^4

Why it's wrong: Dividing the powers does not mean dividing the exponents. The 22 factors on the bottom cancel 22 of the 88 factors on top, so 8−28 - 2 factors stay, not 8÷28 \div 2.

Correct: x8x2=x8−2=x6\frac{x^8}{x^2} = x^{8-2} = x^6

Subtracting in the wrong order: 3732=32−7\frac{3^7}{3^2} = 3^{2-7}

Why it's wrong: The exponent of the numerator always comes first. Here the top has more factors, so after cancelling the factors stay on top and the exponent is positive.

Correct: 3732=37−2=35=243\frac{3^7}{3^2} = 3^{7-2} = 3^5 = 243

Dividing the bases: 6562=13\frac{6^5}{6^2} = 1^3

Why it's wrong: The base tells you which number is multiplied, and it does not change. Only the number of factors changes.

Correct: 6562=65−2=63=216\frac{6^5}{6^2} = 6^{5-2} = 6^3 = 216

Subtracting the coefficients too: 12x74x3=8x4\frac{12x^7}{4x^3} = 8x^4

Why it's wrong: Coefficients are ordinary numbers, and a fraction bar means division: 12÷412 \div 4. Only the exponents of the same base are subtracted.

Correct: 12x74x3=124⋅x7−3=3x4\frac{12x^7}{4x^3} = \frac{12}{4} \cdot x^{7-3} = 3x^4

Interactive Visual

Exponents as repeated factors

Quotient rule

Same base, divide: subtract the exponents.

5652\frac{5^{6}}{5^{2}}

Write each power as repeated factors

55·55·55·55·55·55
55·55

2 common factors cancel, top and bottom

4 factors are left on top

54\displaystyle 5^{4}

Result

5652\displaystyle \frac{5^{6}}{5^{2}}==56−2\displaystyle 5^{6 - 2}==54\displaystyle 5^{4}

Value: 54=6255^{4} = 625

This works for every base except 0

Exponents as repeated factors

Quotient rule

Same base, divide: subtract the exponents.

x9x4\frac{x^{9}}{x^{4}}

Write each power as repeated factors

xx·xx·xx·xx·xx·xx·xx·xx·xx
xx·xx·xx·xx

4 common factors cancel, top and bottom

5 factors are left on top

x5\displaystyle x^{5}

Result

x9x4\displaystyle \frac{x^{9}}{x^{4}}==x9−4\displaystyle x^{9 - 4}==x5\displaystyle x^{5}

This works for every base except 0

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

Practice Problems

18 problems
Problem 1 of 18
Easy
Simplify 7973\frac{7^9}{7^3}.

Why It Matters

Dividing powers comes up whenever numbers are very large or very small:
  • Computers: storage sizes are powers of 2. How many 1 MB files (2202^{20} bytes) fit into 1 GB (2302^{30} bytes)? 230220=210=1024\frac{2^{30}}{2^{20}} = 2^{10} = 1024
  • Science: distances and speeds in scientific notation are divided by subtracting the exponents of 10: 108105=103\frac{10^8}{10^5} = 10^3
  • Mental math: 220217=23=8\frac{2^{20}}{2^{17}} = 2^3 = 8 without ever calculating 220=10485762^{20} = 1048576
  • Algebra: simplifying fractions like 12x74x3\frac{12x^7}{4x^3} is the first step towards rational expressions and solving equations
The quotient rule turns a long division of huge numbers into one small subtraction.

Real World Applications

Photos on a Memory Card

Computer memory is counted in powers of 2, so dividing storage sizes is a job for the quotient rule.

Example:

1 MB is 2202^{20} bytes and 1 KB is 2102^{10} bytes, so 1 MB holds 220210=210=1024\frac{2^{20}}{2^{10}} = 2^{10} = 1024 KB.

1Try It Yourself

A memory card holds 2302^{30} bytes. Each photo takes 2222^{22} bytes.

How many photos fit on the card?

Step 1: Write the mathematical expression

Divide the size of the card by the size of one photo:

How Long Sunlight Needs to Reach Earth

Astronomers divide huge distances by speeds written in scientific notation: divide the numbers in front, subtract the exponents of 10.

Example:

A distance of 6×1086 \times 10^8 km at 3×1053 \times 10^5 km per second takes 63×108−5=2×103=2000\frac{6}{3} \times 10^{8-5} = 2 \times 10^3 = 2000 seconds.

2Try It Yourself

The Sun is about 1.5×1081.5 \times 10^8 km away from Earth. Light travels about 3×1053 \times 10^5 km per second.

About how many seconds does sunlight need to reach Earth?

Step 1: Write the mathematical expression

Time = distance divided by speed:

Budget per Person

Economists divide large amounts of money by the number of people to compare countries of different sizes.

Example:

If 8×1098 \times 10^9 dollars are shared by 4×1064 \times 10^6 people, each share is 2×103=20002 \times 10^3 = 2000 dollars.

3Try It Yourself

A country spends 6×10106 \times 10^{10} dollars on schools in one year and has 2×1072 \times 10^7 inhabitants.

How many dollars per inhabitant is that?

Step 1: Write the mathematical expression

Divide the budget by the number of inhabitants:

Key Takeaways

  • 1Quotient rule: aman=am−n\frac{a^m}{a^n} = a^{m-n} for a≠0a \neq 0: keep the base, subtract the exponents
  • 2The rule works because common factors cancel: each factor on the bottom cancels one factor on top
  • 3Always subtract top minus bottom: 3732=37−2=35\frac{3^7}{3^2} = 3^{7-2} = 3^5
  • 4The base never changes and the exponents are never divided: x8x2=x6\frac{x^8}{x^2} = x^6, not x4x^4
  • 5With coefficients, divide the numbers and subtract the exponents: 12x74x3=3x4\frac{12x^7}{4x^3} = 3x^4
  • 6In a mixed expression, simplify the top with the product rule first, then use the quotient rule

Frequently Asked Questions

Because you cannot divide by zero. With a=0a = 0 the denominator 0n0^n would be 00, so the fraction would have no value at all.
Because you cannot divide by zero. With a=0a = 0 the denominator 0n0^n would be 00, so the fraction would have no value at all.
Then every factor cancels, for example 4343=1\frac{4^3}{4^3} = 1. The rule gives 43−3=404^{3-3} = 4^0, and why 404^0 equals 11 is the topic of the lesson Zero Exponent.
Cancel as usual: 33 factors cancel and 22 factors of 22 stay on the bottom, so 2325=122=14\frac{2^3}{2^5} = \frac{1}{2^2} = \frac{1}{4}. How to write this with a single exponent is shown in the lesson Negative Exponents.
No. The rule only counts factors of one and the same base. Calculate the powers instead: 6423=12968=162\frac{6^4}{2^3} = \frac{1296}{8} = 162.

Glossary

Quotient
The result of a division. In aman\frac{a^m}{a^n} the powers form a quotient.
Base
The number or variable that is multiplied. In 565^6 the base is 55.
Exponent
The number of factors of the base. In 565^6 the exponent is 66.
Common factor
A factor that appears both on top and on the bottom of a fraction, so it cancels to 11.
Coefficient
The number in front of a variable. In 12x712x^7 the coefficient is 1212.

Formula Card

Quotient rule

aman=am−n\frac{a^m}{a^n} = a^{m-n}

Same base $a \neq 0$: keep the base, subtract the exponents (top minus bottom)

With coefficients

c amd an=cd am−n\frac{c\,a^m}{d\,a^n} = \frac{c}{d}\,a^{m-n}

Divide the coefficients, subtract the exponents

Product rule, then quotient rule

am⋅apan=am+p−n\frac{a^m \cdot a^p}{a^n} = a^{m+p-n}

Add the exponents on top, then subtract the exponent on the bottom

Today's challengeOne problem, three tries

More in This Topic

Introduction to Exponents

Intermediate25 min

Laws of Exponents

Intermediate25 min

Square Roots and Cube Roots

Intermediate25 min

Product Rule for Exponents

Intermediate20 min