Lesson 10 1 WarmUp ALGEBRA 1 Simplifying Radicals

Lesson 10 -1 Warm-Up ALGEBRA 1

“Simplifying Radicals” (10 -1) What is a “radical expression”? Radical Expression: an expression that involves a radical (the expression inside or under the radical sign is called the radicand) Example: 2 3 x+3 How can you To simplify a radical expression, remove perfect square factors from the simplify a radical radicand. expression? Rule: Multiplication Property of Square Roots: For every number a ≥ 0 and b ≥ 0: ab = a • b Example: 54 = 9 • 6 = 3 • 6 =3 6 You can simplify radical expressions by rewriting them as a product of perfect square factors and the remaining factors Example: Simplify 192 = 64 • 3 64 is a perfect square and factor of 192. = 64 • 3 Multiplication Property of Square Roots = 8 3 Simplify 64 ALGEBRA 1

“Simplifying Radicals” (10 -1) How can you simplify a radical expression that contains a variable? You can simplify radical expressions containing variables with exponents of 2 or greater. A variable with an even exponent is a perfect square (Examples: n 2 = n; n 4 = n 2; ; n 6 = n 3). Therefore, you can use the Multiplication Property of Square Roots to simplify radical expressions containing variables as well (i. e. perfect square factors of the variable times the rest of the factors). Example: Simplify 45 a 5 = 9 • a 4 • 5 • a = = = 9 • a 4 • 5 • a 3 • a 2 • 5 • a 3 a 2 5 a 9 and a 4 are perfect square factors of 45 a 5. Multiplication Property of Square Roots Simplify 9 a 4 ALGEBRA 1

Simplifying Radicals LESSON 10 -1 Additional Examples Simplify 243 = = = 9 81 • 3 243. 81 is a perfect square and a factor of 243. 3 Use the Multiplication Property of Square Roots. Simplify 81. ALGEBRA 1

Simplifying Radicals LESSON 10 -1 Additional Examples Simplify 28 x 7 = = 4 x 6 • 7 x 4 x 6 • = 2 x 3 7 x 28 x 7. 4 x 6 is a perfect square and a factor of 28 x 7. 7 x Use the Multiplication Property of Square Roots. Simplify 4 x 6. ALGEBRA 1

Simplifying Radicals LESSON 10 -1 Additional Examples Simplify each radical expression. a. 12 • 32 = 12 • 32 Use the Multiplication Property of Square Roots. = 384 Simplify under the radical. = 64 • 6 64 is a perfect square and a factor of 384. = 64 • = 8 6 6 Use the Multiplication Property of Square Roots. Simplify 64. ALGEBRA 1

Simplifying Radicals LESSON 10 -1 Additional Examples (continued) b. 7 5 x • 3 8 x = 21 40 x 2 Multiply the whole numbers and use the Multiplication Property of Square Roots. = 21 4 x 2 • 10 factor of 40 x 2. 4 x 2 is a perfect square and a = 21 4 x 2 • 10 Square Roots. Use the Multiplication Property of = 21 • 2 x Simplify = 42 x 10 10 4 x 2. Simplify. ALGEBRA 1

Simplifying Radicals LESSON 10 -1 Additional Examples Suppose you are looking out a fourth floor window 52 ft above the ground. Use the formula d = 1. 5 h to estimate the distance you can see to the horizon. Round your answer to the nearest mile. d = 1. 5 h = 1. 5 • 52 Substitute 52 for h. = 78 Multiply. 8. 83176 Use a calculator. To the nearest mile, the distance you can see is 9 miles. ALGEBRA 1

“Simplifying Radicals” (10 -1) Rule: Division Property of Square Roots: This says that you van simplify the What is the Division Property radical expressions of the numerator and denominator separately. of Square Roots? For every number a ≥ 0 and b 0: a b = a b Example: ALGEBRA 1

Simplifying Radicals LESSON 10 -1 Additional Examples Simplify each radical expression. a. b. 13 64 13 = 64 13 64 Use the Division Property of Square Roots. = 13 8 Simplify 49 = x 4 49 x 4 Use the Division Property of Square Roots. 64. 49 x 4 = 7 x 2 Simplify 49 and x 4. ALGEBRA 1

“Simplifying Radicals” (10 -1) Tip: When the denominator of a radicand is not a perfect square, it may be easier to divide the numerator by the denominator before simplifying the radicand. Example: ALGEBRA 1

Simplifying Radicals LESSON 10 -1 Additional Examples Simplify each radical expression. a. 120 10 120 = 10 12 Divide. = 4 • 3 4 is a perfect square and a factor of 12. = 4 • =2 3 3 Use the Multiplication Property of Square Roots. Simplify 4. ALGEBRA 1

Simplifying Radicals LESSON 10 -1 Additional Examples (continued) b. 75 x 5 48 x 75 x 5 = 48 x 25 x 4 16 Divide the numerator and denominator by 3 x. = 25 x 4 16 Use the Division Property of Square Roots. = 25 • 16 5 x 2 = 4 x 4 Use the Multiplication Property of Square Roots. Simplify 25, x 4, and 16. ALGEBRA 1

“Simplifying Radicals” (10 -1) What does it mean to “rationalize” the denominator? Rationalize: If the denominator of a radical expression is not a perfect square, it is an irrational number (the square root of any number that is not a perfect square is irrational). To “rationalize” the denominator (make it into a perfect square), multiply both the numerator and denominator by the denominator to create an equal fraction in which the denominator is no longer in radical form. Example: ALGEBRA 1

Simplifying Radicals LESSON 10 -1 Additional Examples Simplify by rationalizing the denominator. a. 3 7 3 = 7 = = √ 7 3 • 7 √ 7 Multiply by 3 7 49 Use the Multiplication Property of Square Roots. 3 7 7 to make the denominator a perfect square. Simplify 49. ALGEBRA 1

Simplifying Radicals LESSON 10 -1 Additional Examples (continued) b. 11 12 x 3 11 = 12 x 3 √ 3 x 11 • 12 x 3 √ 3 x Multiply by 3 x to make the denominator a 3 x perfect square. = 33 x 36 x 4 Use the Multiplication Property of Square Roots. = 33 x 6 x 2 Simplify 36 x 4. ALGEBRA 1

Simplifying Radicals LESSON 10 -1 Lesson Quiz Simplify each radical expression. 1. 16 • 8 8 4. 2 a 5 2 a a 3 2 2. 4 5. 144 3 x 15 x 3 48 3. 12 36 3 3 5 5 x ALGEBRA 1