algebra 1 lesson 10-3 warm-up. algebra 1 “operations with radical expressions” (10-3) what are...

ALGEBRA 1

Lesson 10-3 Warm-Up

ALGEBRA 1

“Operations With Radical Expressions” (10-3)

What are “like and unlike radicals”?

How can you combine like radicals?

like radicals: radical expressions that have the same radicand

Example: 4 7 and -12 7 are like radicals.

unlike radicals: radical expressions that do not have the same radicand

Example: 3 11 and 2 5 are NOT like radicals

You can combine like radicals using the Distributive Property.

Example: Simplify 2 and 3 2 .

2 + 3 2 = 1 2 + 3 2 Both terms contain 2 .

(1 + 3) 2 Use Distributive Property to combine like terms [like 2x + 3x = (2 + 3)x = 5x]

4 2 Simplify.

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= (4 + 1) 3 Use the Distributive Property to combine like radicals.

= 5 3 Simplify.

Simplify 4 3 + 3.

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

Operations With Radical ExpressionsLESSON 10-3

Additional Examples

ALGEBRA 1

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

= 8 5 – 9 • 5 Use the Multiplication Property of Square Roots.

= 8 5 – 3 5 Simplify 9.

= 5 5 Simplify.

Simplify 8 5 – 45.

= (8 – 3) 5 Use the Distributive Property tocombine like terms.

Operations With Radical ExpressionsLESSON 10-3

Additional Examples

ALGEBRA 1

= 4 • 10 + 9 5 Use the Multiplication Property of Square Roots.

= 2 10 + 9 5 Simplify.

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

Simplify 5 ( 8 + 9).

Operations With Radical ExpressionsLESSON 10-3

Additional Examples

ALGEBRA 1

“Operations With Radical Expressions” (10-3)

How do simplify using FOILing?

If both radical expressions have two terms, you can FOIL in the same way you would when multiplying two binomials.

Example:

Given.

ALGEBRA 1

Simplify ( 6 – 3 21)( 6 + 21).

( 6 – 3 21)( 6 + 21)

= 36 +1 126 - 3 126 – 3 441 Use

FOIL.= 6 – 2 126 – 3(21) Combine like radicals 126 and

simplify 36 and 441.

= 6 – 2 9 • 14 – 63 9 is a perfect square factor of 126.

= 6 – 2• 9 • 14 – 63 Use the Multiplication Property of Square Roots.

= 6 – 6 14 – 63 Simplify 9.

= – 57 – 6 14 Simplify.

Operations With Radical ExpressionsLESSON 10-3

Additional Examples

ALGEBRA 1

“Operations With Radical Expressions” (10-3)

What are “conjugates”?

How can we rationalize a denominator using conjugates?

conjugates: The sum and the difference of the same two terms.

Example:

Rule: The product of two conjugates is the difference of two squares.

Example:

FOIL

Simplify.

Notice that the product of two conjugates containing radicals has no radicals.

Recall that a simplified radical expression has no radical in the denominator. If the denominator does contain a radical, we need to get rid of it through rationalization. If the denominator is a sum or difference that contains a radical expression, we can rationalize it by multiplying the numerator and denominator by the conjugate of the denominator.

Example: To rationalize , multiply by

ALGEBRA 1

“Operations With Radical Expressions” (10-3)

Example:

Multiply (the denominator is the sum of the squares)

Divide 6 and 3 by the common factor 3

Simplify.

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Simplify . 8

7 – 3

= • Multiply the numerator and denominator by the conjugate of the denominator.

8

7 – 3

7 + 3

7 + 3

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

= 2 7 + 2 3 Simplify the expression.

= Simplify the denominator. 8( 7 + 3)

4

= Multiply in the denominator. 8( 7 + 3)

7 – 3

Operations With Radical ExpressionsLESSON 10-3

Additional Examples

2

1

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Define: 51 = length of painting x = width of painting

Words: (1 + 5) : 2 = length : width

Translate: =

x (1 + 5) = 102 Cross multiply.

= Solve for x by dividing both side by

(1+ 5).

102

(1 + 5)

x(1 + 5)

(1 + 5)

51 x

(1 + 5) 2

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.

Operations With Radical ExpressionsLESSON 10-3

Additional Examples

ALGEBRA 1

(continued)

x = Multiply in the denominator.102(1 – 5) 1 – 5

x = Simplify the denominator.102(1 – 5) –4

x = Divide 102 and –4 by the common factor –2.

– 51(1 – 5) 2

x = 31.51973343 Use a calculator.

x 32The exact width of the painting is inches.

The approximate width of the painting is 32 inches.

– 51(1 – 5) 2

x = • Multiply the numerator and the denominator by the conjugate of the denominator.

(1 – 5)

(1 – 5) 102

(1 + 5)

Operations With Radical ExpressionsLESSON 10-3

Additional Examples

ALGEBRA 1

16

5 – 7

Simplify each expression.

1. 12 16 – 2 16 2. 20 – 4 5 3. 2( 2 + 3 3)

4. ( 3 – 2 21)( 3 + 3 21) 5.

40 –2 5 2 + 3 6

–123 + 3 7 –8 5 – 8 7

Operations With Radical ExpressionsLESSON 10-3

Lesson Quiz