Learn on PengiSaxon Algebra 1Chapter 8: Advanced Factoring and Functions

Lesson 76: Multiplying Radical Expressions

In this Grade 9 Saxon Algebra 1 lesson, students learn to multiply radical expressions using the Product Property of Radicals, applying techniques such as the Distributive Property and FOIL to simplify expressions involving square roots, including binomials with radicals. The lesson covers multiplying monomials and binomials containing radicals, squaring radical expressions, and simplifying results by combining like terms. Real-world applications, such as calculating the area of a rectangular rug with radical dimensions, reinforce how these skills connect to geometry and measurement.

Section 1

📘 Multiplying Radical Expressions

New Concept

The square root of a product equals the product of the square roots of the factors.

ab=a⋅b where a≥0 and b≥0 \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \text{ where } a \ge 0 \text{ and } b \ge 0

What’s next

Next, you’ll apply this rule to simplify products and multiply more complex expressions involving radicals, like binomials.

Section 2

Product Property of Radicals

Property

The square root of a product equals the product of the square roots of the factors.

ab=a⋅b where a≥0 and b≥0 \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \text{ where } a \ge 0 \text{ and } b \ge 0

Explanation

Think of this as a matchmaking rule! Numbers outside the radicals multiply together, and numbers inside the radicals (radicands) multiply together. You can't mix and match an outsider with an insider. Keep them in their own groups, combine them, and then simplify the result to get your final, perfect couple.

Examples

123=36=6 \sqrt{12}\sqrt{3} = \sqrt{36} = 6
53⋅27=(5⋅2)(3⋅7)=1021 5\sqrt{3} \cdot 2\sqrt{7} = (5 \cdot 2)(\sqrt{3} \cdot \sqrt{7}) = 10\sqrt{21}
(45)2=42⋅(5)2=16⋅5=80 (4\sqrt{5})^2 = 4^2 \cdot (\sqrt{5})^2 = 16 \cdot 5 = 80

Section 3

Example Card: Simplifying Products of Radicals

Multiplying separate radicals looks tricky, but this example on the Product Property of Radicals shows we can combine them easily.

Example Problem
Simplify the expression 35y⋅10y3\sqrt{5y} \cdot \sqrt{10y}.

Step-by-Step

  1. We'll use the Product Property of Radicals, a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}, to combine the radical terms.
35y⋅10y=35y⋅10y 3\sqrt{5y} \cdot \sqrt{10y} = 3\sqrt{5y \cdot 10y}
  1. Multiply the expressions inside the radical.
350y2 3\sqrt{50y^2}
  1. Simplify the radical. The largest perfect square that divides 5050 is 2525. Rewrite the radicand as a product of factors.
325⋅2⋅y2 3\sqrt{25 \cdot 2 \cdot y^2}
  1. Take the square root of the perfect squares (2525 and y2y^2) and move them outside the radical.
3⋅5⋅y2 3 \cdot 5 \cdot y \sqrt{2}
  1. Multiply the coefficients outside the radical for the final answer.
15y2 15y\sqrt{2}

Section 4

Applying the Distributive Property

Property

Use the Distributive Property to multiply a radical by each term inside a parenthesis.

a(b+c)=ab+ac a(b+c) = ab + ac

Explanation

Imagine you are handing out snacks at a party. The radical outside the parentheses is the snack, and you have to give one to every single person (term) inside. Don’t be rude and skip anyone! After you’ve distributed the radical snack to everyone, check to see if any of the new radicals can be simplified.

Examples

3(5+2)=53+6 \sqrt{3}(5 + \sqrt{2}) = 5\sqrt{3} + \sqrt{6}
5(10−5)=50−25=52−5 \sqrt{5}(\sqrt{10} - \sqrt{5}) = \sqrt{50} - \sqrt{25} = 5\sqrt{2} - 5

Section 5

Multiplying Binomials with Radicals

Property

Use the Distributive Property or FOIL method to multiply binomials containing radicals. For squaring a binomial:

(a−b)2=a2−2ab+b2 (a - b)^2 = a^2 - 2ab + b^2

Explanation

Remember your old pal FOIL? (First, Outer, Inner, Last). It’s the ultimate tool for multiplying two binomials, even with tricky radicals. This ensures every term gets multiplied by every other term. Watch out for the classic trap of just squaring the first and last parts of a binomial—that's a no-go!

Examples

(5+2)(3−2)=15−52+32−4=13−22 (5 + \sqrt{2})(3 - \sqrt{2}) = 15 - 5\sqrt{2} + 3\sqrt{2} - \sqrt{4} = 13 - 2\sqrt{2}
(7−5)2=72−2(7)5+(5)2=49−145+5=54−145 (7 - \sqrt{5})^2 = 7^2 - 2(7)\sqrt{5} + (\sqrt{5})^2 = 49 - 14\sqrt{5} + 5 = 54 - 14\sqrt{5}

Section 6

Example Card: Multiplying Binomials with Radicals

You already know how to FOIL binomials. Let's apply that same skill to expressions with radicals, a key idea from this lesson.

Example Problem
Simplify the expression (5+2)(3−8)(5 + \sqrt{2})(3 - \sqrt{8}).

Step-by-Step

  1. Use the FOIL method (First, Outer, Inner, Last) to multiply the binomials.
(5)(3)⏟First+(5)(−8)⏟Outer+(2)(3)⏟Inner+(2)(−8)⏟Last \underbrace{(5)(3)}_{\text{First}} + \underbrace{(5)(-\sqrt{8})}_{\text{Outer}} + \underbrace{(\sqrt{2})(3)}_{\text{Inner}} + \underbrace{(\sqrt{2})(-\sqrt{8})}_{\text{Last}}
  1. Calculate each product.
15−58+32−16 15 - 5\sqrt{8} + 3\sqrt{2} - \sqrt{16}
  1. Simplify any radicals. We can simplify 8\sqrt{8} and 16\sqrt{16}.
8=4⋅2=22 \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}
16=4 \sqrt{16} = 4
  1. Substitute these simplified forms back into the expression.
15−5(22)+32−4 15 - 5(2\sqrt{2}) + 3\sqrt{2} - 4
15−102+32−4 15 - 10\sqrt{2} + 3\sqrt{2} - 4
  1. Combine like terms: group constants and radical terms.
(15−4)+(−102+32)=11−72 (15 - 4) + (-10\sqrt{2} + 3\sqrt{2}) = 11 - 7\sqrt{2}

Book overview

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Chapter 8: Advanced Factoring and Functions

  1. Lesson 1

    Lesson 71: Making and Analyzing Scatter Plots

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  2. Lesson 2

    Lesson 72: Factoring Trinomials: x² + bx + c

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  3. Lesson 3

    Lesson 73: Solving Compound Inequalities

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  4. Lesson 4

    Lesson 74: Solving Absolute-Value Equations

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  5. Lesson 5

    Lesson 75: Factoring Trinomials: ax² + bx + c

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  6. Lesson 6Current

    Lesson 76: Multiplying Radical Expressions

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  7. Lesson 7

    Lesson 77: Solving Two-Step and Multi-Step Inequalities

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  8. Lesson 8

    Lesson 78: Graphing Rational Functions

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  9. Lesson 9

    Lesson 79: Factoring Trinomials by Using the GCF

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  10. Lesson 10

    Lesson 80: Calculating Frequency Distributions

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Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Multiplying Radical Expressions

New Concept

The square root of a product equals the product of the square roots of the factors.

ab=a⋅b where a≥0 and b≥0 \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \text{ where } a \ge 0 \text{ and } b \ge 0

What’s next

Next, you’ll apply this rule to simplify products and multiply more complex expressions involving radicals, like binomials.

Section 2

Product Property of Radicals

Property

The square root of a product equals the product of the square roots of the factors.

ab=a⋅b where a≥0 and b≥0 \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \text{ where } a \ge 0 \text{ and } b \ge 0

Explanation

Think of this as a matchmaking rule! Numbers outside the radicals multiply together, and numbers inside the radicals (radicands) multiply together. You can't mix and match an outsider with an insider. Keep them in their own groups, combine them, and then simplify the result to get your final, perfect couple.

Examples

123=36=6 \sqrt{12}\sqrt{3} = \sqrt{36} = 6
53⋅27=(5⋅2)(3⋅7)=1021 5\sqrt{3} \cdot 2\sqrt{7} = (5 \cdot 2)(\sqrt{3} \cdot \sqrt{7}) = 10\sqrt{21}
(45)2=42⋅(5)2=16⋅5=80 (4\sqrt{5})^2 = 4^2 \cdot (\sqrt{5})^2 = 16 \cdot 5 = 80

Section 3

Example Card: Simplifying Products of Radicals

Multiplying separate radicals looks tricky, but this example on the Product Property of Radicals shows we can combine them easily.

Example Problem
Simplify the expression 35y⋅10y3\sqrt{5y} \cdot \sqrt{10y}.

Step-by-Step

  1. We'll use the Product Property of Radicals, a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}, to combine the radical terms.
35y⋅10y=35y⋅10y 3\sqrt{5y} \cdot \sqrt{10y} = 3\sqrt{5y \cdot 10y}
  1. Multiply the expressions inside the radical.
350y2 3\sqrt{50y^2}
  1. Simplify the radical. The largest perfect square that divides 5050 is 2525. Rewrite the radicand as a product of factors.
325⋅2⋅y2 3\sqrt{25 \cdot 2 \cdot y^2}
  1. Take the square root of the perfect squares (2525 and y2y^2) and move them outside the radical.
3⋅5⋅y2 3 \cdot 5 \cdot y \sqrt{2}
  1. Multiply the coefficients outside the radical for the final answer.
15y2 15y\sqrt{2}

Section 4

Applying the Distributive Property

Property

Use the Distributive Property to multiply a radical by each term inside a parenthesis.

a(b+c)=ab+ac a(b+c) = ab + ac

Explanation

Imagine you are handing out snacks at a party. The radical outside the parentheses is the snack, and you have to give one to every single person (term) inside. Don’t be rude and skip anyone! After you’ve distributed the radical snack to everyone, check to see if any of the new radicals can be simplified.

Examples

3(5+2)=53+6 \sqrt{3}(5 + \sqrt{2}) = 5\sqrt{3} + \sqrt{6}
5(10−5)=50−25=52−5 \sqrt{5}(\sqrt{10} - \sqrt{5}) = \sqrt{50} - \sqrt{25} = 5\sqrt{2} - 5

Section 5

Multiplying Binomials with Radicals

Property

Use the Distributive Property or FOIL method to multiply binomials containing radicals. For squaring a binomial:

(a−b)2=a2−2ab+b2 (a - b)^2 = a^2 - 2ab + b^2

Explanation

Remember your old pal FOIL? (First, Outer, Inner, Last). It’s the ultimate tool for multiplying two binomials, even with tricky radicals. This ensures every term gets multiplied by every other term. Watch out for the classic trap of just squaring the first and last parts of a binomial—that's a no-go!

Examples

(5+2)(3−2)=15−52+32−4=13−22 (5 + \sqrt{2})(3 - \sqrt{2}) = 15 - 5\sqrt{2} + 3\sqrt{2} - \sqrt{4} = 13 - 2\sqrt{2}
(7−5)2=72−2(7)5+(5)2=49−145+5=54−145 (7 - \sqrt{5})^2 = 7^2 - 2(7)\sqrt{5} + (\sqrt{5})^2 = 49 - 14\sqrt{5} + 5 = 54 - 14\sqrt{5}

Section 6

Example Card: Multiplying Binomials with Radicals

You already know how to FOIL binomials. Let's apply that same skill to expressions with radicals, a key idea from this lesson.

Example Problem
Simplify the expression (5+2)(3−8)(5 + \sqrt{2})(3 - \sqrt{8}).

Step-by-Step

  1. Use the FOIL method (First, Outer, Inner, Last) to multiply the binomials.
(5)(3)⏟First+(5)(−8)⏟Outer+(2)(3)⏟Inner+(2)(−8)⏟Last \underbrace{(5)(3)}_{\text{First}} + \underbrace{(5)(-\sqrt{8})}_{\text{Outer}} + \underbrace{(\sqrt{2})(3)}_{\text{Inner}} + \underbrace{(\sqrt{2})(-\sqrt{8})}_{\text{Last}}
  1. Calculate each product.
15−58+32−16 15 - 5\sqrt{8} + 3\sqrt{2} - \sqrt{16}
  1. Simplify any radicals. We can simplify 8\sqrt{8} and 16\sqrt{16}.
8=4⋅2=22 \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}
16=4 \sqrt{16} = 4
  1. Substitute these simplified forms back into the expression.
15−5(22)+32−4 15 - 5(2\sqrt{2}) + 3\sqrt{2} - 4
15−102+32−4 15 - 10\sqrt{2} + 3\sqrt{2} - 4
  1. Combine like terms: group constants and radical terms.
(15−4)+(−102+32)=11−72 (15 - 4) + (-10\sqrt{2} + 3\sqrt{2}) = 11 - 7\sqrt{2}

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 8: Advanced Factoring and Functions

  1. Lesson 1

    Lesson 71: Making and Analyzing Scatter Plots

    LearnPractice
  2. Lesson 2

    Lesson 72: Factoring Trinomials: x² + bx + c

    LearnPractice
  3. Lesson 3

    Lesson 73: Solving Compound Inequalities

    LearnPractice
  4. Lesson 4

    Lesson 74: Solving Absolute-Value Equations

    LearnPractice
  5. Lesson 5

    Lesson 75: Factoring Trinomials: ax² + bx + c

    LearnPractice
  6. Lesson 6Current

    Lesson 76: Multiplying Radical Expressions

    LearnPractice
  7. Lesson 7

    Lesson 77: Solving Two-Step and Multi-Step Inequalities

    LearnPractice
  8. Lesson 8

    Lesson 78: Graphing Rational Functions

    LearnPractice
  9. Lesson 9

    Lesson 79: Factoring Trinomials by Using the GCF

    LearnPractice
  10. Lesson 10

    Lesson 80: Calculating Frequency Distributions

    LearnPractice