Laws of Exponents

Master the fundamental rules for working with exponents: product rule, quotient rule, power rule, and more.

Intermediate25 minLesson
Printable worksheet

Definition

The laws of exponents are rules that simplify calculations involving powers. When working with exponents that have the same base, these laws help us multiply, divide, and raise powers to powers.
The Seven Laws:
LawRuleExample
Product Ruleam×an=am+na^m \times a^n = a^{m+n}23×22=252^3 \times 2^2 = 2^5
Quotient Ruleam÷an=am−na^m \div a^n = a^{m-n}35÷32=333^5 \div 3^2 = 3^3
Power Rule(am)n=am×n(a^m)^n = a^{m \times n}(23)2=26(2^3)^2 = 2^6
Zero Exponenta0=1a^0 = 1 (where a≠0a \neq 0)50=15^0 = 1
Negative Exponenta−n=1ana^{-n} = \frac{1}{a^n}2−3=182^{-3} = \frac{1}{8}
Power of a Product(ab)n=an×bn(ab)^n = a^n \times b^n(2×3)2=4×9(2 \times 3)^2 = 4 \times 9
Power of a Quotient(ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}(23)2=49\left(\frac{2}{3}\right)^2 = \frac{4}{9}

Try it now

Simplify 23×242^3 \times 2^4

Worked Examples

Simplify 34×323^4 \times 3^2
1

Identify the base and exponents

Base: 33, Exponents: 44 and 22 → Same base, so we can use the product rule
2

Apply the product rule

am×an=am+na^m \times a^n = a^{m+n} → 34×32=34+23^4 \times 3^2 = 3^{4+2}
3

Add the exponents

4+2=64 + 2 = 6 → 363^6
4

Verify (optional)

34=813^4 = 81, 32=93^2 = 9, 81×9=72981 \times 9 = 729, 36=7293^6 = 729 ✓ → Confirmed!

Common Mistakes

Adding exponents when multiplying different bases: 23×32=652^3 \times 3^2 = 6^5

Why it's wrong: The product rule only works when the bases are the SAME. 23×32=8×9=722^3 \times 3^2 = 8 \times 9 = 72, not 65=77766^5 = 7776.

Correct: Keep different bases separate: 23×32=8×9=722^3 \times 3^2 = 8 \times 9 = 72

Multiplying exponents when using the product rule: x3×x4=x12x^3 \times x^4 = x^{12}

Why it's wrong: For multiplication, we ADD exponents. Multiplying exponents is for the power rule.

Correct: x3×x4=x3+4=x7x^3 \times x^4 = x^{3+4} = x^7

Thinking a0=0a^0 = 0

Why it's wrong: a0a^0 means no factors of aa at all, and a product with no factors is 1, not 0. Think: anan=an−n=a0=1\frac{a^n}{a^n} = a^{n-n} = a^0 = 1

Correct: a0=1a^0 = 1 for any a≠0a \neq 0

Confusing negative exponents with negative numbers: 2−3=−82^{-3} = -8

Why it's wrong: A negative exponent means reciprocal, not negative value.

Correct: 2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8} (a positive fraction)

Interactive Visual

Exponents as repeated factors

Product rule

Same base, multiply: add the exponents.

34⋅323^{4} \cdot 3^{2}

Write each power as repeated factors

33·33·33·33
343^{4}4 factors
·
33·33
323^{2}2 factors

4 + 2 = 6 factors in all

Result

34⋅32\displaystyle 3^{4} \cdot 3^{2}==34+2\displaystyle 3^{4 + 2}==36\displaystyle 3^{6}

Value: 36=7293^{6} = 729

Exponents as repeated factors

Quotient rule

Same base, divide: subtract the exponents.

5754\frac{5^{7}}{5^{4}}

Write each power as repeated factors

55·55·55·55·55·55·55
55·55·55·55

4 common factors cancel, top and bottom

3 factors are left on top

53\displaystyle 5^{3}

Result

5754\displaystyle \frac{5^{7}}{5^{4}}==57−4\displaystyle 5^{7 - 4}==53\displaystyle 5^{3}

Value: 53=1255^{3} = 125

This works for every base except 0

Exponents as repeated factors

Power of a power

Power of a power: multiply the exponents.

(23)4\left(2^{3}\right)^{4}

Write each power as repeated factors

22·22·22
232^{3}3 factors
22·22·22
232^{3}3 factors
22·22·22
232^{3}3 factors
22·22·22
232^{3}3 factors

4 groups of 3: 4 · 3 = 12 factors

Result

(23)4\displaystyle \left(2^{3}\right)^{4}==23⋅4\displaystyle 2^{3 \cdot 4}==212\displaystyle 2^{12}

Value: 212=40962^{12} = 4096

2³ = 8
3

See how powers of a number grow on the number line. Change the base and exponent.

Interactive Sandbox

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Practice Problems

16 problems
Problem 1 of 16
Easy
Simplify 23×242^3 \times 2^4

Why It Matters

The laws of exponents are essential for:
  • Scientific notation: Scientists write very large or small numbers using powers of 10
  • Computer science: Data storage is measured in powers of 2 (KB, MB, GB)
  • Finance: Compound interest calculations use exponential growth
  • Physics: Exponential decay describes radioactive materials
Without these laws, simplifying expressions like x5×x3x2\frac{x^5 \times x^3}{x^2} would require writing out all the factors!

Real World Applications

Computer Memory and Storage

Computer storage is measured in powers of 2. Understanding exponent laws helps calculate total storage.

Example:

1 KB is 2102^{10} bytes and 1 MB is 2102^{10} KB, so 1 MB is 210×210=2202^{10} \times 2^{10} = 2^{20} bytes.

1Try It Yourself

A server has 8 memory modules, each with 2302^{30} bytes (1 GB).

Write the total memory as a single power of 2.

Step 1: Write the mathematical expression

Total = 8×2308 \times 2^{30}. Since 8=238 = 2^3:

Exponents in Chemistry and Biology

Scientists use exponent laws to work with very large or small numbers.

Example:

Avogadro's number is about 6×10236 \times 10^{23}. For 2 moles: 2×6×1023=12×1023=1.2×101×1023=1.2×10242 \times 6 \times 10^{23} = 12 \times 10^{23} = 1.2 \times 10^{1} \times 10^{23} = 1.2 \times 10^{24} particles (product rule: 101×1023=102410^1 \times 10^{23} = 10^{24}).

2Try It Yourself

A sample contains 10810^8 bacteria. Every bacterium splits into 2, and this happens 3 times.

How many bacteria are there after the 3 divisions?

Step 1: Write the mathematical expression

Starting number times growth factor:

Key Takeaways

  • 1Product Rule: am×an=am+na^m \times a^n = a^{m+n} (add exponents when multiplying same base)
  • 2Quotient Rule: am÷an=am−na^m \div a^n = a^{m-n} (subtract exponents when dividing same base)
  • 3Power Rule: (am)n=am×n(a^m)^n = a^{m \times n} (multiply exponents for power of a power)
  • 4Zero Exponent: a0=1a^0 = 1 (any non-zero number to the power 0 equals 1)
  • 5Negative Exponent: a−n=1ana^{-n} = \frac{1}{a^n} (negative exponent means reciprocal)
  • 6The product and quotient rules need the same base: 23×322^3 \times 3^2 cannot be written as one power

Frequently Asked Questions

Consider a3a3\frac{a^3}{a^3}. By the quotient rule: a3−3=a0a^{3-3} = a^0. But any number divided by itself equals 1. So a0=1a^0 = 1.
Consider a3a3\frac{a^3}{a^3}. By the quotient rule: a3−3=a0a^{3-3} = a^0. But any number divided by itself equals 1. So a0=1a^0 = 1.
No! The product and quotient rules only work when the bases are identical. 23×322^3 \times 3^2 cannot be simplified to a single power.
Fractional exponents represent roots. For example, a1/2=aa^{1/2} = \sqrt{a}. The same laws apply: a1/2×a1/2=a1=aa^{1/2} \times a^{1/2} = a^1 = a.

Glossary

Base
The number being raised to a power. In 535^3, the base is 5.
Exponent
The power to which a base is raised. In 535^3, the exponent is 3.
Power
The result of raising a base to an exponent. 53=1255^3 = 125, so 125 is a power of 5.
Product Rule
When multiplying powers with the same base, add the exponents: am×an=am+na^m \times a^n = a^{m+n}
Quotient Rule
When dividing powers with the same base, subtract the exponents: am÷an=am−na^m \div a^n = a^{m-n}
Reciprocal
The multiplicative inverse. The reciprocal of aa is 1a\frac{1}{a}.
Today's challengeOne problem, three tries

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Product Rule for Exponents

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Quotient Rule for Exponents

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