Rewriting Radical Expressions

Key Concepts

• Rewriting radical expressions
• Product property of square roots
• Multiplying radical expressions

Introduction 

• Radical expression: 

An expression containing a square root is called a radical expression. 

• Radicand: 

The number or variable inside the radical symbol is called the radicand. 

Rewriting Radical Expression 

Rewriting radical expression means simplifying the radical expression to its simplest form or until the radical is removed, if feasible. 

Rewriting the radical expressions in algebra means reducing the radical expressions until the radical is removed, if possible. 

Product Property 

Product property of radical states that the square root of the product of numbers is equal to the product of square roots of each of the numbers.  

Let us see some examples to understand this. 

Example 1: 

Rewrite the expression √40 by using product property.  

Solution: 

Step1: Given expression √40..… (1) 

Step2: Now we will use the product property for equation (1), 

√40 = √5 x 8

= √5 × √8

= √5× √2*2*2

= √5 × 2√2

= 2√5 x 2

= 2√10

The simplest form of the radical expression √ 40 is 2√10. 

Relation between Side length and Area 

Let us see the relationship between side length, area of a square, and role of radicals in it. 

Example: 

Suppose the area of a square as 36 sq.cm, find the side length. 

Solution: 

Area of a square is 36 sq.cm 

Now we need to find the side length, 

Side length S = √Area

S=√36

=√2×2×3×3

=2×3

=6 cm

Visualizing Radical Expressions 

Let us see the visualization of radical expressions. 

How can you show √36 is equivalent to √29 ? 

Let us see this. 

The sum of lengths of two smaller orange lines is equal to the length of the larger orange line. 

Given a graph as shown in the image below, find the factored form of the quadratic equation. 

Comparing Radical Expressions 

• Can we compare radical expressions? 

Yes,  

Let us see some examples 

Example 1: 

Compare √ 84 to 2√3√7.

Solution: 

Consider √84 

= √12×7 

= √2×2×3×7 

= 2√21 

= 2√3√7 

Hence both the radical expressions are equivalent. 

Example 2: 

Compare √ 36 to 3√6.

Solution: 

Consider √36

 = √12×3 

= √2×2×3×3 

= 2×3

= 6

= 3√6=√3×2

= 3√3√2

Hence, both the radical expressions are not equivalent. 

Example 3: 

What is an equivalent expression for √182 .

Solution: 

Consider √182

= √2×7×13 

= √2√7√13

This radical expression cannot be reduced further. 

The radical expression=√2√7√13 is equivalent to √182.

Radical Expressions with Variables 

Example: 

What is an equivalent expression for √72×5  .

Solution: 

Consider √72×5 

= √2×2×2×3×3×x×x×x×x×x 

= √2²×2 ×3²×x²×x²×x 

= 2×3×x×x√2×x

= 6x²√2x 

This radical expression cannot be reduced further. 

The radical expression √72×5  is equivalent to 6x²√2x .

Multiplying Radical Expressions 

Can we multiply radical expressions? 

Yes

Let us see an example. 

Rewrite the radical expression 2√32x⁶ 5√48x⁷ to the simplest form.

Solution:

2√32x⁶ x 5√48x⁷

= 2 x 5√32x⁶ × 48x⁷

= 10√2²× 2²× 2 × x² × x² × x² × 2² × 2² × 3 × x² × x² × 2 × x

= 10 × 2 × 2 × x × x × x × 2 × 2 × x × x × x√2 × 3 × x

= 160x⁶ √6x

The radical expression 2√32x⁶ 5/48x⁷ is equivalent to 160x⁶ √6x

Constructing Radical Expressions (Real life examples) 

Can we write a radical expression for the data given? 

Yes,  

Let us see an example. 

Example 1: 

Suppose Smitha’s house is 3 m to the north of play store and Raju’s house is 4 m to the west of play store. How far is Smitha’s house from Raju’s house? 

Solution: 

By the Pythagorean theorem, we get  

Distance =√3²+4²

We need to simplify the radical expression √3²+4² to get the distance between Raju’s house and Smitha’s house. 

√3²+4² = √9+16 =√25 =5 m

Example 2: 

A car skidded m ft during a race. The expression s = √9m is the rate of speed of the car in ft/s. Find the rate of the speed of the car from the given image. 

Solution: 

Rate of speed s = √9m ft/s 

= √9×196

= √3×3×14×14

= 3×14

= 42ft/s

Exercise

1. Rewrite the expression√48 by using product property.
2. Suppose the area of a square is 84 sq.cm, find the side length.
3. A car skidded m ft during a race. The expression s = √8m is the rate of speed of the car in ft/s. Find the rate of the speed of the car, if the car skidded 169 ft.
4. Suppose a school is 12 m to the north of Toy store and Jack’s office is 5 m to the west of Toy store. How far is the school from Jack’s office?
5. Rewrite the radical expression 2√45t6 x 3√121×7 to the simplest form.
6. What is an equivalent expression for √125×5.
7. How can you show √24 is equivalent to 2√6.
8. Compare √16 to 2√4
9. Rewrite √98 into simplest form.
10. Compare √108 to 9√2.

Concept Map

What have we learned

• Rewriting radical expressions
• Product property of square roots
• Multiplying radical expressions