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Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 6

Multiply, Divide, and Raise Powers

Use the product, quotient, and power rules. Commute coefficients first, then combine matching bases.

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Key Ideas Easy Medium Hard Super Challenge

x² × x³ = x⁵

(x²)³ = x⁶

What to Remember

Product rule

Same base, multiply: add the exponents.

am × an = am+n

x² × x³ = x⁵

Quotient rule

Same base, divide: subtract the exponents.

aman = am−n

x⁵x² = x³

Power rule

A power raised to a power: multiply the exponents.

(am)n = amn

(x²)³ = x⁶

Coefficients: commute first

Use the commutative property to put the numbers together and the matching bases together, then simplify.

6x³ × 2x⁴ = 6 × 2 × x³ × x⁴

Level 1

Easy

One rule. Keep the same base.

Example
Simplify. x³ × x⁵

Same base, so add the exponents.

x3+5 = x⁸

Answer: x⁸

1
Simplify. y² × y⁶
Show Answer

Same base, so add the exponents.

y2+6 = y⁸

Answer: y⁸

2
Simplify. a⁴ × a
Show Answer

a = a¹

Add the exponents.

a4+1 = a⁵

Answer: a⁵

3
Simplify. m⁹m³
Show Answer

Same base, so subtract the exponents.

m9−3 = m⁶

Answer: m⁶

4
Simplify. (w²)⁴
Show Answer

Power of a power: multiply the exponents.

w2 × 4 = w⁸

Answer: w⁸

5
Simplify. 4³ × 4²
Show Answer

Same base, so add the exponents.

43+2 = 4⁵

Answer: 4⁵

Level 2

Medium

Use the commutative property: numbers together, matching bases together.

Example
Simplify. 6x³ × 2x⁴

Use the commutative property to rearrange: numbers together, matching bases together.

6x³ × 2x⁴ = 6 × 2 × x³ × x⁴

6 × 2 = 12

x³ × x⁴ = x3+4 = x⁷

12x⁷

Answer: 12x⁷

1
Simplify. 5p² × 3p⁶
Show Answer

Use the commutative property to rearrange.

5 × 3 = 15

p² × p⁶ = p⁸

15p⁸

Answer: 15p⁸

2
Simplify. 12n⁹4n²
Show Answer

Rearrange: numbers together, matching bases together.

124reduce = 3

n⁹n² = n⁷

3n⁷

Answer: 3n⁷

3
Simplify. 8t⁸2t³
Show Answer

82reduce = 4

t⁸t³ = t⁵

4t⁵

Answer: 4t⁵

4
Simplify. (3x²)³
Show Answer

The outer exponent hits every factor.

(3x²)³ = 3³ × (x²)³

3³ = 27

(x²)³ = x⁶

27x⁶

Answer: 27x⁶

5
Simplify. 2a³b × 5ab⁴
Show Answer

Use the commutative property to rearrange.

2 × 5 = 10

a³ × a = a⁴

b × b⁴ = b⁵

10a⁴b⁵

Answer: 10a⁴b⁵

Level 3

Hard

Use more than one rule. Leave the answer with positive exponents.

Example
Simplify. (x²)³ × x⁴x⁵

(x²)³ = x⁶

Use the commutative property: matching bases together.

x⁶ × x⁴x⁵ = x¹⁰x⁵ = x⁵

Answer: x⁵

1
Simplify. (2y³)² × 3y
Show Answer

(2y³)² = 4y⁶

4 × 3 = 12

y⁶ × y = y⁷

12y⁷

Answer: 12y⁷

2
Simplify. 18a⁵b³6a²b
Show Answer

Rearrange: numbers together, matching bases together.

186reduce = 3

a⁵a² = a³

b³b = b²

3a³b²

Answer: 3a³b²

3
Simplify. (m⁴n)³ × mn²
Show Answer

(m⁴n)³ = m¹²n³

m¹² × m = m¹³

n³ × n² = n⁵

m¹³n⁵

Answer: m¹³n⁵

4
Simplify. (3x²)²9x
Show Answer

(3x²)² = 9x⁴

9x⁴9x = x³

Answer: x³

5
Simplify. 4x⁰ × x⁵
Show Answer

x⁰ = 1 if x ≠ 0

4 × 1 × x⁵ = 4x⁵

Answer: 4x⁵

Super Challenge

Super Challenge

1
Show two ways that (x³)⁴ × x² = x¹⁴: expand as a product of fourteen x’s in words, and use the laws.
Show Answer

Power rule: (x³)⁴ = x¹². Then x¹² × x² = x¹⁴.

A product of four groups of three x’s is twelve x’s, then two more is fourteen x’s.

Answer: x¹⁴

2
A student writes x⁸x² = x⁴ because 8 ×12 = 4. What rule did they use by mistake?
Show Answer

They multiplied instead of subtracting.

Quotient rule: x8−2 = x⁶.

Answer: x⁶

3
Fill in the missing exponent. a³ × ak = a¹¹. Then check with k = 8.
Show Answer

3 + k = 11, so k = 8.

Check: a³ × a⁸ = a¹¹.

Answer: k = 8

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