Lesson 10-3 Warm-Up

“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.

= (4 + 1) 3 Use the Distributive Property to combine like radicals. Operations With Radical Expressions LESSON 10-3 Additional Examples Simplify 4 3 + 3. 4 3 + 3 = 4 3 + 1 3 Both terms contain 3. = (4 + 1) 3 Use the Distributive Property to combine like radicals. = 5 3 Simplify.

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 8 5 – 45. 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. = (8 – 3) 5 Use the Distributive Property to combine like terms. = 5 5 Simplify.

5( 8 + 9) = 40 + 9 5 Use the Distributive Property. Operations With Radical Expressions LESSON 10-3 Additional Examples Simplify 5 ( 8 + 9). 5( 8 + 9) = 40 + 9 5 Use the Distributive Property. = 4 • 10 + 9 5 Use the Multiplication Property of Square Roots. = 2 10 + 9 5 Simplify.

“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.

= 6 – 2 126 – 3(21) Combine like radicals and simplify 36 and 441. Operations With Radical Expressions LESSON 10-3 Additional Examples 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 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 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

“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.

= • 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 7 + 3 = Multiply in the denominator. 8( 7 + 3) 7 – 3 = Simplify the denominator. 8( 7 + 3) 4 = 2( 7 + 3) Divide 8 and 4 by the common factor 4. = 2 7 + 2 3 Simplify the expression.

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 (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. 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) 51 x 2

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 (1 + 5) 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 32 The exact width of the painting is inches. The approximate width of the painting is 32 inches. – 51(1 – 5) 2

Simplify each expression. Operations With Radical Expressions LESSON 10-3 Lesson Quiz 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