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

Exponent Laws: Powers of Powers, Products and Quotients, and Simplifying Expressions

B2.2 A1About 75 minutes

Learning goals

I can explain and use the power of a power law, (aᵐ)ⁿ = aᵐⁿ.
I can explain and use the power of a product and power of a quotient laws, (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ.
I can combine all the exponent laws to simplify numeric and algebraic expressions, writing answers with positive exponents.
I can simplify an expression first and then evaluate it, and check my work by substituting numbers.

Words to know

power of a power law
(aᵐ)ⁿ = aᵐⁿ: keep the base and multiply the exponents.

power of a product law
(ab)ⁿ = aⁿbⁿ: every factor inside the brackets is raised to the exponent.

power of a quotient law
(a/b)ⁿ = aⁿ/bⁿ, b ≠ 0: the numerator and denominator are both raised to the exponent.

simplest form (for powers)
Each base appears once, there are no brackets left to expand, and all exponents are positive.

Ice cubes and exponents

A cube-shaped ice block has side length 3x² cm. Its volume is (3x²)³. What is that as a simple monomial? You need two new laws: one for raising a power to a power, and one for raising a product to a power. By the end of this lesson you will be able to say (3x²)³ = 27x⁶ in one step and explain why.

Power of a power

(x²)³ means x² × x² × x². By the product law, that is x²⁺²⁺² = x⁶. Adding 2 three times is the same as 2 × 3.

(aᵐ)ⁿ = aᵐⁿ (keep the base, multiply the exponents)

(x²)³ is three groups of x · x. That is 2 × 3 = 6 factors of x, so (x²)³ = x⁶.

Power of a product and power of a quotient

(2a)³ = (2a)(2a)(2a). Multiplication can be done in any order, so regroup: (2 · 2 · 2)(a · a · a) = 2³a³ = 8a³.

(ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ (b ≠ 0)

Every factor inside the brackets gets the exponent, including the coefficient and any negative sign.

(2a)³ has three 2s and three a's. Regrouping them shows (2a)³ = 2³a³ = 8a³. A common error is to write 2a³, which forgets to cube the 2.

Law	Rule	Example
Product	aᵐ × aⁿ = aᵐ⁺ⁿ	x³ × x⁴ = x⁷
Quotient	aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0)	x⁹ ÷ x² = x⁷
Power of a power	(aᵐ)ⁿ = aᵐⁿ	(x³)⁴ = x¹²
Power of a product	(ab)ⁿ = aⁿbⁿ	(3y)² = 9y²
Power of a quotient	(a/b)ⁿ = aⁿ/bⁿ (b ≠ 0)	(x/2)³ = x³/8
Zero exponent	a⁰ = 1 (a ≠ 0)	(5xy)⁰ = 1
Negative exponent	a⁻ⁿ = 1/aⁿ (a ≠ 0)	x⁻² = 1/x²

All the exponent laws in one place. Each one comes from writing powers in expanded form.

Worked example

Example 1: Power of a power and of a product

Simplify: a) (5²)³ b) (−2x³y)⁴ c) (3a²/b³)²

a) (5²)³ = 5⁶ = 15 625.
b) Every factor gets the exponent 4: (−2)⁴ (x³)⁴ y⁴ = 16x¹²y⁴. (Even exponent, so the sign is positive.)
c) Numerator and denominator: 3²(a²)² / (b³)² = 9a⁴/b⁶.

15 625; 16x¹²y⁴; 9a⁴/b⁶

Worked example

Example 2: Combine several laws

Simplify (2x²)³(3x)².

Brackets with exponents first: (2x²)³ = 8x⁶ and (3x)² = 9x².
Multiply: 8 × 9 = 72 and x⁶ × x² = x⁸.

72x⁸

Worked example

Example 3: Quotients and negative exponents

Simplify (x⁴y²)(x³y) ÷ (x²y⁵). Write the answer with positive exponents.

Numerator: x⁴⁺³ y²⁺¹ = x⁷y³.
Divide: x⁷⁻² y³⁻⁵ = x⁵y⁻².
y⁻² = 1/y², so the answer is x⁵/y².

x⁵/y²

Worked example

Example 4: A negative exponent outside brackets

Simplify (2a⁻³)⁻².

Every factor gets the exponent −2: 2⁻² (a⁻³)⁻² = 2⁻² a⁶.
2⁻² = 1/4.

a⁶/4

Worked example

Example 5: Simplify, then evaluate

Evaluate (2x²y)³ ÷ (4x⁴y²) when x = 3 and y = −2.

Simplify: (2x²y)³ = 8x⁶y³. Then 8x⁶y³ ÷ 4x⁴y² = 2x²y.
Substitute: 2(3)²(−2) = 2 × 9 × (−2) = −36.
Check without simplifying: (2 × 9 × (−2))³ ÷ (4 × 81 × 4) = (−36)³ ÷ 1296 = −46 656 ÷ 1296 = −36. ✓

−36. Simplifying first made the arithmetic much easier.

Watch out!

(3x)² = 9x², but 3x² = 3 · x². The exponent only applies to what it touches, unless there are brackets.
(−2x)² = 4x² (positive), but −(2x)² = −4x².
(a + b)² is not a² + b². Test with a = 3, b = 4: (3 + 4)² = 49, but 3² + 4² = 25. The power of a product law is for products, not sums. You will learn how to expand (a + b)² correctly later in high school math.

Hands-on activity

Doubling a cube

You need: 27 small identical items that stack into cubes, such as sugar cubes, dice or building bricks. If you use small items, keep them away from young children.

Build a cube with side length 1 item. Volume = 1³ = 1.
Build a cube with side length 2. Count the items: 2³ = 8.
Build a cube with side length 3: 3³ = 27.
Now think algebraically. If a cube has side s, its volume is s³. If you double the side, the volume is (2s)³ = 2³s³ = 8s³: 8 times as big. Check with steps 1 and 2: 1 → 8.
Predict: if you triple the side, how many times bigger is the volume? ((3s)³ = 27s³, so 27 times. Check with steps 1 and 3.)

Talk about it: A pizza with double the diameter has (2r)² = 4r² for the r² part of its area, so 4 times as much pizza!

Just for fun

Why are exponents so confident?

Because whatever you put them on, they raise it to a higher power.

Your turn

Simplify (x⁵)².

Simplify (4m³)².

Simplify (−3a²b)³.

Simplify (2x/y²)⁴.

Simplify (x²)³ × x⁴.

Simplify (6a⁵b³)² ÷ (9a⁴b⁶).

Simplify (3x⁻²)² with positive exponents.

Find the volume of a cube with side 3x² cm.

More practice: warm-up

Simplify (2³)² and evaluate.

Simplify (ab)⁵.

Simplify (y/3)².

Simplify (7x)⁰.

Simplify (k⁴)³.

More practice: on your own

Simplify (−2a³)⁴(a²)³.

Simplify (10x⁴y)² ÷ (5x³y²).

Simplify (p⁻²q³)⁻¹ with positive exponents.

Evaluate (3a²)² ÷ (a³) when a = −2, by simplifying first.

A square has side length 5ab² m. Write its area as a monomial.

Challenge

Write 4³ × 8² as a single power of 2.

Simplify (2x³y⁻¹)³ ÷ (4x²y⁻⁵) with positive exponents.

Kai says (x³)² = x³² and Leah says (x³)² = x⁹. Who is right? Explain the error each person made.

Check your understanding

Quiz

Simplify (a⁴)³.

Simplify (−2y)³.

Simplify (x/5)².

Simplify (2x³)²(x⁴).

Simplify (3m⁻²)⁻¹.

Which statement is false?

For parents and teachers

Big idea: This lesson completes expectation B2.2: students analyse the relationships between exponents and operations, and use all the exponent laws to simplify numeric and algebraic expressions. These skills are used constantly in Grade 10 math (expanding, factoring) and in science (units like m/s²).

Common difficulties:

Not applying the exponent to the coefficient: (2a)³ written as 2a³.
Adding instead of multiplying exponents for a power of a power.
Leaving negative exponents in the final answer. Ask for positive exponents.
Distributing an exponent over a sum. (a + b)² ≠ a² + b².

How to help: After your teen simplifies an expression, ask them to pick a small number for each variable (like 2) and check that the original and the simplified expression give the same value. This builds the A1 habit of checking work.