Grade by Grade
B. Exponential Functions · Exponential Functions

The Exponent Rules: Multiplying, Dividing and a Power of a Power

B1.5About 70 minutes
Learning goals
  • I can explain, using expanded form and patterns, why aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ and (aᵐ)ⁿ = aᵐⁿ.
  • I can use the exponent rules to simplify numerical expressions with integer, fraction and negative bases.
  • I can tell when the exponent rules apply (same base, multiplying or dividing) and when they do not (sums, different bases).
  • I can simplify first and then evaluate, and check my answer with a calculator.

Words to know

power
An expression such as 2⁵ that means repeated multiplication. 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.
base
The number being multiplied in a power. In (−3)⁴ the base is −3; in −3⁴ the base is 3.
exponent
The small raised number that tells how many times the base is used as a factor.
product rule
To multiply powers with the same base, keep the base and add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ.
quotient rule
To divide powers with the same base, keep the base and subtract the exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0).
power of a power rule
To raise a power to an exponent, keep the base and multiply the exponents: (aᵐ)ⁿ = aᵐⁿ.

The snow-day phone tree

A big storm is coming and the school board cancels buses. The principal calls 3 families. Each of those families calls 3 more, and so on. After 2 rounds of calls, 3 × 3 = 3² = 9 families hear the news in round 2. Each of those 9 families now starts its own little tree that runs for 3 more rounds. Each little tree reaches 3³ = 27 families in its last round.

So round 5 has 3² × 3³ families. But round 5 is just five rounds of tripling, so it is also 3⁵ = 243. That means 3² × 3³ = 3⁵. The exponents added. That is not a coincidence, and in this lesson you will see exactly why.

The whole Exponential Functions unit runs on powers, so we start by making the exponent rules automatic.

Remember from Grades 9 and 10

In Grade 9 you learned the product, quotient and power of a power laws and also (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ. In Grade 10 you used patterns to show that a⁰ = 1 and a⁻ⁿ = 1/aⁿ, and you compared the graphs of y = x² and y = 2ˣ. In this course you will push these ideas further: fraction bases, rational (fraction) exponents, and full exponential functions that model growth and decay.

Why the product rule works

A power is a short way to write repeated multiplication. To multiply two powers with the same base, write each one out in expanded form and count the factors:

2³ × 2² = (2 × 2 × 2) × (2 × 2) = 2 × 2 × 2 × 2 × 2 = 2⁵

Three factors plus two factors gives five factors. In general:

aᵐ × aⁿ = aᵐ⁺ⁿ

The base stays the same. You do not multiply the bases. 2³ × 2² is 2⁵ = 32, not 4⁵ = 1024.

2³ × 2² means222×223 factors2 factors=222223 + 2 = 5 factors= 2⁵ = 32
Multiplying powers with the same base joins the two strings of factors. Count them: 3 + 2 = 5.

Why the quotient rule works

Division undoes multiplication, so common factors cancel:

3⁵ ÷ 3² = (3 × 3 × 3 × 3 × 3) ÷ (3 × 3) = 3 × 3 × 3 = 3³

Two factors on the bottom cancel two factors on top, so 5 − 2 = 3 are left:

aᵐ ÷ aⁿ = aᵐ⁻ⁿ, as long as a ≠ 0 (you can never divide by 0).

3⁵ ÷ 3² written as a fraction:3333333=3 × 3 × 3= 3³ = 27Each 3 on the bottom cancels one 3 on top: 5 − 2 = 3 factors are left.
In a quotient, every factor on the bottom cancels one on top. Subtract the exponents: 5 − 2 = 3.

Why the power of a power rule works

(5³)² means 5³ used as a factor twice: 5³ × 5³. By the product rule that is 5³⁺³ = 5⁶. Two groups of three factors gives 2 × 3 = 6 factors:

(aᵐ)ⁿ = aᵐⁿ

A quick way to remember all three: multiplying powers → add exponents; dividing powers → subtract exponents; a power of a power → multiply exponents. Each rule is one step "down" from the operation you see.

(5³)² means 5³ × 5³5555³5555³= 2 groups of 3 factors = 5⁶Raising a power to a power multiplies the exponents: 3 × 2 = 6.
(5³)² is two copies of 5³, so there are 2 × 3 = 6 factors of 5.

Investigating with successive quotients

You can also discover the rules from a table. Look at powers of 2 and divide each value by the one before it:

The values 2, 4, 8, 16, 32, 64 each get multiplied by 2 when the exponent goes up by 1. So going up 2 in the exponent multiplies by 2 × 2 = 2², going up 3 multiplies by 2³, and so on. That is exactly the product rule: 2⁴ × 2² means "start at 2⁴ and go up 2 more steps", landing on 2⁶.

Power2¹2²2³2⁴2⁵2⁶
Value248163264
Value ÷ previous value22222
Successive quotients are all 2: each step up in the exponent multiplies by the base.
Worked example

Example 1: A fraction base

Simplify (1/2)³ × (1/2)², then evaluate.

  1. The bases are the same (1/2), and we are multiplying, so add the exponents: (1/2)³⁺² = (1/2)⁵.
  2. Evaluate: (1/2)⁵ = 1⁵/2⁵ = 1/32.
  3. Check by expanding: (1/2)³ = 1/8 and (1/2)² = 1/4. (1/8) × (1/4) = 1/32. ✓

(1/2)⁵ = 1/32 (or 0.031 25).

Worked example

Example 2: Use all three rules

Simplify (3⁴ × 3⁵) ÷ (3²)³, then evaluate.

  1. Top: product rule. 3⁴ × 3⁵ = 3⁴⁺⁵ = 3⁹.
  2. Bottom: power of a power. (3²)³ = 3⁶, because 2 × 3 = 6.
  3. Divide: quotient rule. 3⁹ ÷ 3⁶ = 3⁹⁻⁶ = 3³.
  4. Evaluate: 3³ = 27. (Check on a calculator: 3⁹ = 19 683 and 3⁶ = 729; 19 683 ÷ 729 = 27. ✓)

3³ = 27

Worked example

Example 3: A negative base

Simplify [(−2)⁵ × (−2)³] ÷ (−2)⁶ and evaluate.

  1. Same base, −2, the whole time. The brackets show the negative sign is part of the base.
  2. Top: (−2)⁵⁺³ = (−2)⁸. Then (−2)⁸ ÷ (−2)⁶ = (−2)².
  3. (−2)² = (−2)(−2) = 4. An even number of negative factors gives a positive answer.

(−2)² = 4

When the rules do NOT apply

  • Different bases: 2³ × 5² cannot be written as one power of 2 or of 5. Just evaluate: 8 × 25 = 200.
  • Sums and differences: 2³ + 2² is not 2⁵. It is 8 + 4 = 12. The rules are only for multiplying and dividing.
  • Watch the base: −3⁴ means −(3⁴) = −81, but (−3)⁴ = 81.
  • Do not multiply the base by the exponent: 2⁵ is 32, not 10.
Just for fun

Why are exponents such good motivational speakers?

Because they always tell the base to rise above it.

Hands-on activity

Build a phone tree on paper

You need: a large sheet of paper, a pencil, and a calculator.

  1. Draw one dot at the top (the principal). Draw 3 lines down to 3 dots (round 1). From each of those, draw 3 more lines and dots (round 2). Count the round-2 dots: you should have 3² = 9.
  2. Pick just one round-2 dot. Continue its tree for 3 more rounds on a corner of the page, or just count with your calculator: it reaches 3, then 9, then 27 dots.
  3. All 9 round-2 dots do the same, so round 5 has 9 × 27 dots. Multiply on your calculator.
  4. Now compute 3⁵ on the calculator. Compare.
  5. Change the rule: each family calls 2 others. Predict the number in round 7 using 2³ × 2⁴, then check by doubling seven times.

What to look for: 9 × 27 = 243 = 3⁵, so 3² × 3³ = 3⁵. With 2 calls each, round 7 has 2⁷ = 128 families, and 2³ × 2⁴ = 8 × 16 = 128 too. The tree grows very fast, which is the big idea of this whole unit.

Your turn

1

Write as a single power: 7⁴ × 7⁶

2

Write as a single power: 10⁹ ÷ 10⁵

3

Write as a single power: (4³)⁵

4

Simplify, then evaluate: (2/3)² × (2/3)

5

Simplify, then evaluate: (−5)⁷ ÷ (−5)⁴

6

Simplify, then evaluate: (6² × 6⁵) ÷ 6⁶

7

Simplify, then evaluate: (2⁴)³ ÷ 2¹⁰

8

Is 3⁴ + 3⁴ equal to 3⁸? Explain.

More practice: warm-up

1

Name the base and exponent in (−4)³.

2

Write as a single power: 5² × 5³ × 5

3

Write as a single power: 9⁸ ÷ 9

4

Write as a single power: (3²)⁴

5

Evaluate: 2³ × 2²

More practice: on your own

1

Simplify, then evaluate: (1/3)⁶ ÷ (1/3)⁴

2

Simplify, then evaluate: [(0.5)³]²

3

Simplify, then evaluate: (4⁵ × 4²) ÷ (4³)²

4

Simplify, then evaluate: [(−3)⁴ × (−3)⁵] ÷ [(−3)²]³

5

Find the error: "5³ × 5⁴ = 25⁷".

Challenge

1

Find n: 2ⁿ × 2³ = 2¹¹.

2

Write 8³ × 4² as a single power of 2.

3

A phone tree triples each round. Round 4 reached 81 families. Explain, using the quotient rule, how many times as many families round 10 reaches compared with round 4.

Check your understanding

Quiz

1.

Which expression equals 6⁴ × 6³?

2.

What is (2⁵)² as a single power?

3.

Evaluate (1/2)⁷ ÷ (1/2)⁴.

4.

Which statement is true?

5.

Simplify (5³ × 5⁶) ÷ (5²)⁴.

6.

Why does the quotient rule require a ≠ 0?

For parents and teachers

Big idea (B1.5): students determine the exponent rules for multiplying and dividing powers and for a power of a power, and use them to simplify numerical expressions. Much of this is review from Grade 9, so it should feel familiar; the new part is using fraction and negative bases confidently.

How to help: Ask your teen to say why a rule works ("three 2s times two 2s is five 2s") instead of just quoting it. Have them check a few answers by expanding or with a calculator.

Common difficulties:

  • Multiplying the bases: 5³ × 5⁴ written as 25⁷. Keep the base.
  • Using the rules on sums: 2³ + 2² is not 2⁵.
  • Losing the negative sign: (−3)⁴ = 81 but −3⁴ = −81. Brackets decide what the base is.
  • Forgetting a hidden exponent of 1: 9⁸ ÷ 9 = 9⁷.

A scientific calculator with a power key (xʸ, yˣ or ^) is all that is needed for this unit.