Algebra1 Radical Expressions

Algebra1RadicalExpressions CONFIDENTIAL

Warm Up Graph each data set. Which kind of model best describes the data? 1) {(-3, 16), (-2, 8), (0, 2), (1, 1), (3, 0.25)} 2) {(-5, 15), (-2, -6), (0, -10), (3, -1), (4, 6)} CONFIDENTIAL

Radical Expressions An expression that contains a radical sign (√) is a radical expression . There are many different types of radical expressions, but in this course, you will only study radical expressions that contain square roots. Examples of radical expressions: 14 l2 + w2 2gd d 5√2 18 4 The expression under a radical sign is the radicand . A radicand may contain numbers, variables, or both. It may contain one term or more than one term. CONFIDENTIAL

Simplest Form of a Square-Root Expression An expression containing square roots is in simplest form when • the radicand has no perfect square factors other than 1. • the radicand has no fractions. • there are no square roots in any denominator. CONFIDENTIAL

Remember that positive numbers have two square roots, one positive and one negative. However, √1 indicates a non-negative square root. When you simplify, be sure that your answer is not negative. To simplify √x2 , you should write √x2 = |1| , because you do not know whether x is positive or negative. Below are some simplified square-root expressions: √x2 = |x| √x3 = x√x √x4 = x2 √x5 = x2√x √x6 = |x3| CONFIDENTIAL

Simplifying Square-Root Expressions Simplify each expression. A) 2 = 1 = 1 72 36 6 B) 32 + 42 = 9 + 16 = 25 = 5 C) x2 + 8x + 16 = (x + 4)2 = |x + 4| CONFIDENTIAL

1a) 256 4 1b) 40 + 9 1c) 52 + 122 1d) (3 - x)2 Now you try! Simplify each expression. CONFIDENTIAL

Product Property of Square Roots WORDS For any nonnegative real numbers a and b, the square root of ab is equal to the square root of a times the square root of b. NUMBERS ALGEBRA CONFIDENTIAL

A) 18 = 9(2) = 9 (2) = 3 (2) B) x4y3 = x4 (y3) = x4 y2 y = x2yy Using the Product Property of Square Roots Simplify. All variables represent nonnegative numbers. Factor the radicand using perfect squares. Product Property of Square Roots Simplify. Product Property of Square Roots Product Property of Square Roots Since y is nonnegative, √y2 = y. CONFIDENTIAL

Now you try! Simplify. All variables represent nonnegative numbers. 2a) 128 2b) x3 y2 2c) 48a2b CONFIDENTIAL

Quotient Property of Square Roots WORDS For any real numbers a and b (a ≥ 0 and b > 0) , the square root of a is equal to the b square root of a divided by the square root of b. NUMBERS ALGEBRA CONFIDENTIAL

A) 5 = 5 9 9 = 5 3 Quotient Property of Square Roots. Simplify. B) a5 = a4 81a 81 = a4 81 = a2 9 Using the Quotient Property of Square Roots Simplify. All variables represent nonnegative numbers. Simplify. Quotient Property of Square Roots. Simplify. CONFIDENTIAL

3a) 12 27 3b) 36 x4 3c) y6 4 Now you try! Simplify. All variables represent nonnegative numbers. CONFIDENTIAL

a) 80 25 = 80 25 = 16(5) 25 = 16 (5) 25 = 4 (5) 5 Using the Product and Quotient Properties Together Simplify. All variables represent nonnegative numbers. Quotient Property Write 80 as 16 (5) . Product Property Simplify. CONFIDENTIAL

b) 4x5 9 = 4x5 9 = 4(x5) 9 = 4 (x4) (x) 9 = 4x4 (5) 3 Quotient Property Write 80 as 16 (5) . Product Property Simplify. CONFIDENTIAL

4a) 20 49 4b) z5 25y2 4c) p6 q10 Now you try! Simplify. All variables represent nonnegative numbers. CONFIDENTIAL

c = a2 + b2 c = 902 + 902 Sports Application A baseball diamond is a square with sides of 90 feet. How far is a throw from third base to first base? Give the answer as a radical expression in simplest form. Then estimate the length to the nearest tenth of a foot. The distance from third base to first base is the hypotenuse of a right triangle. Use the Pythagorean Theorem: c2 = a2 + b2 Solve for c. Substitute 90 for a and b. CONFIDENTIAL

c = 8100 + 8100 c = 16,200 c = 100(81)(2) c = 100 (81) (2) c = 10 (9) (2) c = 90 (2) c ≈ 127.3 Simplify. Factor 16,200 using perfect squares. Use the Product Property of Square Roots. Simplify. Use a calculator and round to the nearest tenth. The distance is 90√2 , or about 127.3, feet. CONFIDENTIAL

Now you try! 5) A softball diamond is a square with sides of 60 feet. How long is a throw from third base to first base in softball? Give the answer as a radical expression in simplest form. Then estimate the length to the nearest tenth of a foot. CONFIDENTIAL

Assessment 1) In the expression, 3x - 6 + 7, what is the radicand ? 2) Your boat is traveling due north from a dock. Your friend’s boat left at the same time from the same dock and is headed due east. After an hour, your friend calls and tells you that he has just stopped because of engine trouble. How far must you travel to meet your friend? Give your answer as a radical expression in simplest form. Then estimate the distance to the nearest mile. CONFIDENTIAL

Simplify. All variables represent nonnegative numbers. 3) 81 4) 98 2 5) (a + 7)2 6) 180 CONFIDENTIAL

7) 17 25 8) 7 16 9) 108 49 10) 204 25 Graph each square-root function. CONFIDENTIAL

Let’s review Radical Expressions An expression that contains a radical sign (√) is a radical expression . There are many different types of radical expressions, but in this course, you will only study radical expressions that contain square roots. Examples of radical expressions: 14 l2 + w2 2gd d 5√2 18 4 The expression under a radical sign is the radicand . A radicand may contain numbers, variables, or both. It may contain one term or more than one term. CONFIDENTIAL