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### Operations w/ Radicals

Chapter 10, Section 3

Targets

- I will be able to simplify sums and differences of radical expressions.
- I will be able to simplify products and quotients of radical expressions.

Simplify sums & differences of radical expressions

- We use the distributive property to combine like terms. Ex) 5x + 4x = (5 + 4)x = 9x
- We will use this same principle to combine like radicals.
- Sometimes, you will need to simplify the radical expression to see if you have like radicals

4 3 + 3 = 4 3 + 1 3Both terms contain 3.

= (4 + 1) 3Use the Distributive Property to

combine like radicals.

= 5 3Simplify.

10-3

8 5 – 45 = 8 5 + 9 • 59 is a perfect square and a factor of 45.

= 8 5 – 9 • 5Use the Multiplication Property of

Square Roots.

= 8 5 – 3 5Simplify 9.

= (8 – 3) 5Use the Distributive Property to

combine like terms.

= 5 5Simplify.

10-3

You Try

- Got It, Prb 1, pg 613
- Got It, Prb 2, pg 614

Simplify products of radical expressions

- Use distributive property or FOIL to multiply (or use the Punnett Square model).

5( 8 + 9) = 40 + 9 5Use the Distributive Property.

= 4 • 10 + 9 5 Use the Multiplication Property

of Square Roots.

= 2 10 + 9 5Simplify.

10-3

You Try

- Got It, Prb 3, pg 614

Simplify quotients of radical expressions

- To rationalizethedenominator in a radical expression (i.e. remove the radicals in the denominator), we multiply numerator & denominator by the conjugate of the denominator.
- Conjugates are the sum and difference of the same two terms.
- Example:

7 – 3

Simplify .

8

7 – 3

7 + 3

7 + 3

=• Multiply the numerator and

denominator by the conjugate

of the denominator.

8( 7 + 3)

7 – 3

8( 7 + 3)

4

= Multiply in the denominator.

= Simplify the denominator.

= 2( 7 + 3)Divide 8 and 4 by the common

factor 4.

= 2 7 + 2 3Simplify the expression.

10-3

Define:51 = length of painting

x = width of painting

Relate: (1 + 5) : 2 = length : width

(1 + 5)

2

Write: =

x (1 + 5) = 102Cross multiply.

= Solve for x.

51

x

102

(1 + 5)

x(1 + 5)

(1 + 5)

A painting has a length : width ratio approximately equal to the golden ratio (1 + 5 ) : 2. The length of the painting is 51 in. Find the exact width of the painting in simplest radical form. Then find the approximate width to the nearest inch.

10-3

(1 + 5)

(1 – 5)

(1 – 5)

x =• Multiply the numerator and the

denominator by the conjugate

of the denominator.

102(1 – 5)

1 – 5

x = Multiply in the denominator.

102(1 – 5)

–4

x =Simplify the denominator.

– 51(1 – 5)

2

– 51(1 – 5)

2

x = Divide 102 and –4 by the

common factor –2.

x = 31.51973343Use a calculator.

x 32

The exact width of the painting is inches.

The approximate width of the painting is 32 inches.

(continued)

10-3

You Try

- Got Its, Prbs 4 & 5, pgs 615-616
- Lesson Check, prbs 1-8, pg 616