1. Lesson 10-3 Warm-Up

2. “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 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 = Both terms contain 2 . (1 + 3) 2 Use Distributive Property to combine like terms [like 2x + 3x = (2 + 3)x = 5x] 4 2 Simplify.

3. = (4 + 1) 3 Use the Distributive Property to combine like radicals.
Operations With Radical Expressions
LESSON 10-3
Additional Examples
Simplify
= Both terms contain
= (4 + 1) 3 Use the Distributive Property to
combine like radicals.
= Simplify.

4. 8 5 – 45 = 8 5 – 9 • 5 9 is a perfect square and a factor of 45.
Operations With Radical Expressions
LESSON 10-3
Additional Examples
Simplify –
– = – • 5 9 is a perfect square and a factor of 45.
= – 9 • 5 Use the Multiplication Property of
Square Roots.
= – Simplify
= (8 – 3) 5 Use the Distributive Property to
combine like terms.
= Simplify.

5. 5( 8 + 9) = 40 + 9 5 Use the Distributive Property.
Operations With Radical Expressions
LESSON 10-3
Additional Examples
Simplify ( ).
5( ) = Use the Distributive Property.
= 4 • Use the Multiplication Property
of Square Roots.
= Simplify.

6. “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.

7. = 6 – 2 126 – 3(21) Combine like radicals and simplify 36 and 441.
Operations With Radical Expressions
LESSON 10-3
Additional Examples
Simplify ( 6 – )( ).
( 6 – )( )
= – Use FOIL.
= 6 – – 3(21) Combine like radicals and
simplify and
= 6 – • 14 – 63 9 is a perfect square factor of 126.
= 6 – • – 63 Use the Multiplication Property of
Square Roots.
= 6 – – 63 Simplify
= – 57 – Simplify.

8. “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. 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

9. “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.

10. = • Multiply the numerator and denominator by the conjugate
Operations With Radical Expressions
LESSON 10-3
Additional Examples
Simplify . 8
7 – 3
= • Multiply the numerator and
denominator by the conjugate
of the denominator. 8
7 – 3
= Multiply in the denominator.
8( )
7 – 3
= Simplify the denominator.
8( ) 4
= 2( ) Divide 8 and 4 by the common
factor 4.
= Simplify the expression.

11. Define: 51 = length of painting x = width of painting
Operations With Radical Expressions
LESSON 10-3
Additional Examples
A painting has a length : width ratio approximately equal to the golden ratio ( ) : 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.
Define: 51 = length of painting
x = width of painting
Words: ( ) : 2 = length : width
Translate: =
x ( ) = 102 Cross multiply.
= Solve for x by dividing both side by ( ).
102
( )
x( ) 51 x 2

12. x = • Multiply the numerator and the denominator by the conjugate
Operations With Radical Expressions
LESSON 10-3
Additional Examples
(continued)
x = • Multiply the numerator and the
denominator by the conjugate
of the denominator.
(1 – 5)
102
( )
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 = Use a calculator.
x 32
The exact width of the painting is inches.
The approximate width of the painting is 32 inches.
– 51(1 – 5) 2

13. Simplify each expression.
Operations With Radical Expressions
LESSON 10-3
Lesson Quiz 16
5 – 7
Simplify each expression.
– – ( )
4. ( 3 – )( ) 5. 40
–2 5 –
– –