Grade by Grade
B. Number · Number: Number Sets, Powers and Rational Numbers in Action

Exponent Laws: Multiplying and Dividing Powers

B2.2 A1About 70 minutes
Learning goals
  • I can use patterns and expanded form to explain why aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
  • I can use the product and quotient laws to simplify and evaluate numeric expressions.
  • I can simplify algebraic expressions such as (4a³b)(−5a²b⁴) and 18x⁷ ÷ 6x².
  • I can spot common errors, such as multiplying the bases or using the laws on sums.

Words to know

product law (product of powers)
When multiplying powers with the same base, keep the base and add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ.
quotient law (quotient of powers)
When dividing powers with the same base, keep the base and subtract the exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0).
expanded form
A power written as repeated multiplication: x⁴ = x · x · x · x.
monomial
A single term made of a number and variables with whole-number exponents, like −5a²b⁴.
coefficient
The number part of a term. In −5a²b⁴ the coefficient is −5.

A shortcut you can prove

How much is 2³ × 2⁴? You could work out 8 × 16 = 128. Or you could notice:

2³ × 2⁴ = (2 · 2 · 2) × (2 · 2 · 2 · 2) = 2⁷

There are 3 + 4 = 7 factors of 2 in total. That is the product law, and it works for any base, including variables. Exponent laws let you simplify long expressions before you calculate, which is how scientists, engineers and computer programmers handle huge and tiny numbers.

2³222×2⁴2222=22222222⁷ (3 + 4 = 7 factors)2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128
Multiplying powers with the same base joins the factors together. 3 factors of 2 and 4 more factors of 2 make 7 factors of 2.

The product law

aᵐ × aⁿ = aᵐ⁺ⁿ (same base: keep the base, add the exponents)

It also works with zero and negative exponents: 5⁶ × 5⁻⁴ = 5⁶⁺⁽⁻⁴⁾ = 5². Check: 5⁶ × 1/5⁴ = 5⁶ ÷ 5⁴ = 5 × 5 = 25 = 5². ✓

The quotient law

aᵐ ÷ aⁿ = aᵐ⁻ⁿ, for a ≠ 0 (same base: keep the base, subtract the exponents)

Dividing cancels pairs of matching factors, because x ÷ x = 1.

x·x·x·x·xx·x=x·x·x= x³Each x ÷ x = 1, so 2 pairs cancel: x⁵ ÷ x² = x⁵⁻² = x³
In x⁵ ÷ x², two factors of x in the numerator cancel with the two in the denominator, leaving 5 − 2 = 3 factors: x³.

The quotient law explains zero and negative exponents

  • x⁴ ÷ x⁴ = 1 (anything non-zero divided by itself). The law gives x⁴⁻⁴ = x⁰. So x⁰ = 1, just like the pattern in the Powers lesson.
  • x² ÷ x⁵ = (x · x) ÷ (x · x · x · x · x) = 1/x³. The law gives x²⁻⁵ = x⁻³. So x⁻³ = 1/x³.

The laws and the patterns agree, which is why mathematicians define zero and negative exponents this way.

Worked example

Example 1: Numeric expressions

Simplify, then evaluate: a) 3⁴ × 3² b) 7⁸ ÷ 7⁶ c) (−2)³ × (−2)⁴ d) 10⁻³ × 10⁵ ÷ 10⁻¹

  1. a) 3⁴⁺² = 3⁶ = 729.
  2. b) 7⁸⁻⁶ = 7² = 49.
  3. c) The base is −2 in both: (−2)³⁺⁴ = (−2)⁷ = −128 (odd exponent, so negative).
  4. d) Work left to right: 10⁻³⁺⁵ = 10², then 10² ÷ 10⁻¹ = 10²⁻⁽⁻¹⁾ = 10³ = 1000.

729, 49, −128, 1000

Worked example

Example 2: Monomials with coefficients

Simplify (4a³b)(−5a²b⁴).

  1. Multiply the coefficients: 4 × (−5) = −20.
  2. Multiply powers of a: a³ × a² = a⁵.
  3. Multiply powers of b: b¹ × b⁴ = b⁵ (b means b¹).

−20a⁵b⁵

Worked example

Example 3: Dividing monomials

Simplify (18x⁷y³) ÷ (−6x²y).

  1. Divide the coefficients: 18 ÷ (−6) = −3.
  2. x⁷ ÷ x² = x⁵.
  3. y³ ÷ y¹ = y².

−3x⁵y²

Worked example

Example 4: Simplify first, then substitute

Evaluate (2x³)(3x²) ÷ x⁴ when x = −5.

  1. Simplify: 2 × 3 = 6; x³ × x² = x⁵; x⁵ ÷ x⁴ = x¹. The expression is 6x.
  2. Substitute: 6(−5) = −30.

−30. Simplifying first avoids computing (−5)⁵ = −3125.

MistakeWhy it is wrongCorrect
2³ × 2⁴ = 4⁷Do not multiply the bases.2⁷
3² × 3⁴ = 3⁸Add exponents when multiplying; do not multiply them.3⁶
2³ × 5² = 10⁵The bases are different, so the law does not apply.8 × 25 = 200
2³ + 2⁴ = 2⁷The laws are for multiplying and dividing, not adding.8 + 16 = 24
x⁶ ÷ x² = x³Subtract exponents; do not divide them.x⁴
Five common exponent-law mistakes. Test any "rule" with small numbers to check it.
Hands-on activity

Rule detective

You need: a calculator, paper, a pencil.

  1. Test the product law with three pairs of your own, such as 2⁵ × 2³ and 2⁸. Do they match?
  2. Test the quotient law with three pairs, such as 3⁷ ÷ 3⁴ and 3³.
  3. Now test each false rule from the table above with numbers. Write a counterexample for each one.
  4. Create one true and one false "rule" of your own and challenge someone at home to find out which is false.

Model answer: 2⁵ × 2³ = 32 × 8 = 256 = 2⁸ ✓. Counterexample for 2³ + 2⁴ = 2⁷: 8 + 16 = 24, but 2⁷ = 128.

Just for fun

Why did 2³ and 2⁴ make such a great team?

When they multiplied, they just added up.

Your turn

1

Simplify 5³ × 5⁶ as a single power.

2

Simplify x⁸ ÷ x³.

3

Simplify 4⁻² × 4⁵, then evaluate.

4

Simplify y² ÷ y⁶. Write the answer with a positive exponent.

5

Simplify (3m²)(7m⁵).

6

Simplify (−24a⁶b²) ÷ (8a²b²).

7

Is 2⁴ × 3⁴ = 6⁸ true? Explain.

More practice: warm-up

1

Simplify a² × a³.

2

Simplify 10⁶ ÷ 10².

3

Simplify 6⁰ × 6⁴.

4

Simplify k × k⁹.

5

Evaluate 2⁹ ÷ 2⁶.

More practice: on your own

1

Simplify (−6x³y)(−2xy⁴).

2

Simplify (15p⁴q⁷) ÷ (−5pq³).

3

Simplify 3⁵ × 3⁻² ÷ 3⁴ and evaluate.

4

A rectangle is 4x³ cm long and 2x² cm wide. Write its area as a monomial, then find it when x = 2.

5

Evaluate (5a⁴)(2a³) ÷ (10a⁶) when a = −7.

Challenge

1

Find n: 2ⁿ × 2⁴ = 2¹⁰.

2

Find n: 5⁷ ÷ 5ⁿ = 1/25.

3

Write 8 × 2⁵ × 4⁻¹ as a single power of 2.

Check your understanding

Quiz

1.

Simplify x⁴ × x⁵.

2.

Simplify m¹² ÷ m⁴.

3.

Simplify (2a²)(−3a⁵).

4.

What is 3⁻⁴ × 3⁶?

5.

Which statement is true?

6.

Simplify (20x⁶y²) ÷ (4x²y⁵).

For parents and teachers

Big idea: Expectation B2.2 asks students to analyse, through patterning, how exponents relate to operations with powers, and to use these relationships to simplify numeric and algebraic expressions. Writing powers in expanded form is the key to understanding (and remembering) every law.

Common difficulties:

  • Multiplying the bases or the exponents (2³ × 2⁴ = 4⁷ or 2¹²).
  • Using the laws on sums (2³ + 2⁴ = 2⁷). The laws are only for multiplication and division.
  • Forgetting the invisible exponent 1 on a variable like b.
  • Sign errors with coefficients. Multiply or divide the numbers separately, using integer sign rules.

How to help: When your teen is unsure, ask them to write the powers in expanded form and count the factors. A "test it with small numbers" habit catches almost every mistake.