Learn on PengienVision, Algebra 1Chapter 9: Solving Quadratic Equations

Lesson 3: Rewriting Radical Expressions

In this Grade 11 Algebra 1 lesson from enVision Chapter 9, students learn how to rewrite radical expressions by applying the Product Property of Square Roots to remove perfect square factors from the radicand. The lesson covers simplifying expressions like the square root of 63 into equivalent forms such as 3 times the square root of 7, and extends to radical expressions containing variable terms with odd exponents. Students also practice multiplying radical expressions and simplifying the results into forms with no perfect square factors remaining.

Section 1

Simplified Square Root

Property

$\sqrt{a}$ is considered simplified if $a$ has no perfect square factors.

Examples

$\sqrt{31}$ is simplified because 31 has no perfect square factors.

$\sqrt{32}$ is not simplified because $32 = 16 \cdot 2$ , and $16$ is a perfect square. The simplified form is $4\sqrt{2}$ .

Section 2

Product Property of Square Roots

Property

If $a, b$ are non-negative real numbers, then $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ .

Simplify a square root using the product property.

Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
Use the product rule to rewrite the radical as the product of two radicals.
Simplify the square root of the perfect square.

Examples

To simplify $\sqrt{48}$ : Find the largest perfect square factor, which is 16. Rewrite as $\sqrt{16 \cdot 3}$ , which becomes $\sqrt{16} \cdot \sqrt{3}$ , simplifying to $4\sqrt{3}$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Simplified Square Root

Property

$\sqrt{a}$ is considered simplified if $a$ has no perfect square factors.

Examples

$\sqrt{31}$ is simplified because 31 has no perfect square factors.

$\sqrt{32}$ is not simplified because $32 = 16 \cdot 2$ , and $16$ is a perfect square. The simplified form is $4\sqrt{2}$ .

Section 2

Product Property of Square Roots

Property

If $a, b$ are non-negative real numbers, then $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ .

Simplify a square root using the product property.

Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
Use the product rule to rewrite the radical as the product of two radicals.
Simplify the square root of the perfect square.

Examples

To simplify $\sqrt{48}$ : Find the largest perfect square factor, which is 16. Rewrite as $\sqrt{16 \cdot 3}$ , which becomes $\sqrt{16} \cdot \sqrt{3}$ , simplifying to $4\sqrt{3}$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Simplified Square Root

Property

$\sqrt{a}$ is considered simplified if $a$ has no perfect square factors.

Examples

$\sqrt{31}$ is simplified because 31 has no perfect square factors.

$\sqrt{32}$ is not simplified because $32 = 16 \cdot 2$ , and $16$ is a perfect square. The simplified form is $4\sqrt{2}$ .

Section 2

Product Property of Square Roots

Property

If $a, b$ are non-negative real numbers, then $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ .

Simplify a square root using the product property.

Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
Use the product rule to rewrite the radical as the product of two radicals.
Simplify the square root of the perfect square.

Examples

To simplify $\sqrt{48}$ : Find the largest perfect square factor, which is 16. Rewrite as $\sqrt{16 \cdot 3}$ , which becomes $\sqrt{16} \cdot \sqrt{3}$ , simplifying to $4\sqrt{3}$ .