CONFIDENTIAL 1

Algebra1Algebra1

Adding and Adding and SubtractingSubtracting

Radical ExpressionsRadical Expressions

CONFIDENTIAL 2

Warm UpWarm Up

1) exponential 2) quadratic

1) y = √(4x – 2)

2) y = -2√(x + 3)

3) y = 1 + √(x + 6)

Find the domain of each square-root function.

CONFIDENTIAL 3

Adding and Subtracting Radical ExpressionsAdding and Subtracting Radical Expressions

Square-root expressions with the same radicand are examples of like radicals .

Like radicals can be combined by adding or subtracting. You can use the Distributive Property to show how this is done:

Notice that you can combine like radicals by adding or subtracting the numbers multiplied by the radical and

keeping the radical the same.

2√4 + 4√5 = √5(2 + 5) = 7√5

6√x - 2√x = √x(6 - 2) = 4√x

CONFIDENTIAL 4

Adding and Subtracting Square-Root ExpressionsAdding and Subtracting Square-Root Expressions

A) 3√7 + 8√7 = √7(3 + 8) = 11√7

B) 9√y - √y = √y(9 - 1) = 8√y

C) 12√2 - 4√11 = √x(6 - 2) = 4√x

D) -8√(3d) + 6√(2d) + 10√(3d)= √(3d)(10 - 8) + 6√(2d) = 2√(3d) + 6√(2d)

Add or subtract.

√y = √y. The terms are like radicals.

The terms are like radicals.

The terms are unlike radicals. Do notcombine.

Identify like radicals.

Combine like radicals.

CONFIDENTIAL 5

Now you try!

1a) -√71b) 3√31c) 8√n1d) √(2s) + 8√(5s)

Add or subtract.

1a) 5√7 - 6√7

1b) 8√3 - 5√3

1c) 4√n + 4√n

1d) √(2s) - √(5s) + 9 √(5s)

CONFIDENTIAL 6

Sometimes radicals do not appear to be like until they are simplified. Simplify allradicals in an expression before trying to

identify like radicals.

CONFIDENTIAL 7

Simplifying Before Adding or SubtractingSimplifying Before Adding or Subtracting

A) √(12) + √(27)

= √{(4)(3)} + √{(9)(3)}

= √4√3 + √9√3

= 2√3 + 3√3

= √3(2 + 3)

= 5√3

Add or subtract.

Factor the radicands using perfect squares.

Product Property of Square Roots

Simplify.

Combine like radicals.

CONFIDENTIAL 8

B) 3√8 + √(45)

= 3√{(4)(2)} + √{(9)(5)}

= 3√4√2 + √9√5

= 3(2)√2 + 3√5

= 6√2 + 3√5

Factor the radicands using perfect squares.

Product Property of Square Roots

Simplify.

The terms are unlike radicals. Do not combine.

CONFIDENTIAL 9

C) 5√(28x) - 8√(7x)

= 5√{(4)(7x)} - 8√(7x)

= 5√4√(7x) - 8√(7x)

= 5(2)√(7x) - 8 √(7x)

= 10√(7x) - 8 √(7x)

= 2√(7x)

Factor 28x using a perfect square.

Product Property of Square Roots

Simplify.

Combine like radicals.

CONFIDENTIAL 10

D) √(125b) + 3√(20b) - √(45b)

= √{(25)(5b)} + 3√{(4)(5b)} - √{(9)(5b)}

= √(25)√(5b) + 3√4√(5b) - √9√(5b)

= 5√(5b) + 3(2)√(5b) - 3√(5b)

= 5√(5b) +6√(5b) - 3√(5b)

= 8√(5b)

Factor the radicands using perfect squares.

Product Property of Square Roots

Simplify.

Combine like radicals.

CONFIDENTIAL 11

Now you try!

2a) 5√62b) 9√3 2c) 5√(3y)

Add or subtract.

2a) √(54) + √(24)

2b) 4√(27) - √(18)

2c) √(12y) + √(27y)

CONFIDENTIAL 12

Geometry ApplicationGeometry Application

A) 12 + 5√7 + √(28)

= 12 + 5√7 + √{(4)(7)}

= 12 + 5√7 + √4√7

= 12 + 5√7 + 2√7

= 12 + 7√7

Write an expression for perimeter.

Product Property of Square Roots

Simplify.

Find the perimeter of the triangle. Give your answer as a radical expression in simplest form.

Combine like radicals.

Factor 28 using a perfect square.

The perimeter is (12 + 7√7) cm.

CONFIDENTIAL 13

Now you try!

3 )10√b in.

3) Find the perimeter of a rectangle whose length is 2√b inches and whose width is 3√b inches. Give your answer

as a radical expression in simplest form.

CONFIDENTIAL 14

Assessment

1) 14√3 - 6√3

2) 9√5 + √5

3) 6√2 + 5√2 - 15√2

1 )8√32 )10√53- )4√2

CONFIDENTIAL 15

Simplify each expression. 4 )2√25 )17√36 )33√3

7( - )√5x)8 )2(√7c+)18√(6c)

9 )8(√2t-)4√(3t)

4) √(32) - √(8)

5) 4√(12) + √(75)

6) 2√3 + 5√(12) - 15√(27)

7) √(20x) - √(45x)

8) √(28c) + 9√(24c)

9) √(50t) - 2√(12t) + 3√(2t)

CONFIDENTIAL 16

10) Find the perimeter of the trapezoid shown. Give your answer as a radical expression in simplest form.

10) 12√2 in

CONFIDENTIAL 17

Adding and Subtracting Radical ExpressionsAdding and Subtracting Radical Expressions

Square-root expressions with the same radicand are examples of like radicals .

Like radicals can be combined by adding or subtracting. You can use the Distributive Property to show how this is done:

Notice that you can combine like radicals by adding or subtracting the numbers multiplied by the radical and

keeping the radical the same.

2√4 + 4√5 = √5(2 + 5) = 7√5

6√x - 2√x = √x(6 - 2) = 4√x

Let’s review

CONFIDENTIAL 18

Adding and Subtracting Square-Root ExpressionsAdding and Subtracting Square-Root Expressions

A) 3√7 + 8√7 = √7(3 + 8) = 11√7

B) 9√y - √y = √y(9 - 1) = 8√y

C) 12√2 - 4√11 = √x(6 - 2) = 4√x

D) -8√(3d) + 6√(2d) + 10√(3d)= √(3d)(10 - 8) + 6√(2d) = 2√(3d) + 6√(2d)

Add or subtract.

√y = √y. The terms are like radicals.

The terms are like radicals.

The terms are unlike radicals. Do notcombine.

Identify like radicals.

Combine like radicals.

CONFIDENTIAL 19

Sometimes radicals do not appear to be like until they are simplified. Simplify allradicals in an expression before trying to

identify like radicals.

CONFIDENTIAL 20

Simplifying Before Adding or SubtractingSimplifying Before Adding or Subtracting

A) √(12) + √(27)

= √{(4)(3)} + √{(9)(3)}

= √4√3 + √9√3

= 2√3 + 3√3

= √3(2 + 3)

= 5√3

Add or subtract.

Factor the radicands using perfect squares.

Product Property of Square Roots

Simplify.

Combine like radicals.

CONFIDENTIAL 21

C) 5√(28x) - 8√(7x)

= 5√{(4)(7x)} - 8√(7x)

= 5√4√(7x) - 8√(7x)

= 5(2)√(7x) - 8 √(7x)

= 10√(7x) - 8 √(7x)

= 2√(7x)

Factor 28x using a perfect square.

Product Property of Square Roots

Simplify.

Combine like radicals.

CONFIDENTIAL 22

Geometry ApplicationGeometry Application

A) 12 + 5√7 + √(28)

= 12 + 5√7 + √{(4)(7)}

= 12 + 5√7 + √4√7

= 12 + 5√7 + 2√7

= 12 + 7√7

Write an expression for perimeter.

Product Property of Square Roots

Simplify.

Find the perimeter of the triangle. Give your answer as a radical expression in simplest form.

Combine like radicals.

Factor 28 using a perfect square.

The perimeter is (12 + 7√7) cm.

CONFIDENTIAL 23

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