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Section 7.1

Section 7.1. Roots and Radical Expressions. Write each number as a square of a number. 1. 25 2. 0.09 3. Write each expression as a square of an expression. 4. x 10 5. x 4 y 2 6. 169 x 6 y 12. 4 49. Roots and Radical Expressions. ALGEBRA 2 LESSON 7-1. (For help, go to Lesson 5-4.).

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Section 7.1

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  1. Section 7.1 Roots and Radical Expressions

  2. Write each number as a square of a number. 1. 25 2. 0.09 3. Write each expression as a square of an expression. 4.x105.x4y26. 169x6y12 4 49 Roots and Radical Expressions ALGEBRA 2 LESSON 7-1 (For help, go to Lesson 5-4.) Check Skills You’ll Need 7-1

  3. 1. 25 = 522. 0.09 = 0.323. = = 2 4.x10 = (x5)25.x4y2 = (x2y)26. 169x6y12 = (13x3y6)2 4 49 22 72 2 7 Roots and Radical Expressions ALGEBRA 2 LESSON 7-1 Solutions 7-1

  4. 1 64 1 64 1 64 a. the cube root of 0.027, –125, and 1 4 1 4 Since 3 = , is the cube root of . 81 625 81 625 81 625 81 625 b. the fourth roots of 625, –0.0016, and –3 5 3 5 3 5 3 5 Since 4 = and 4 = , and – are fourth roots of . Roots and Radical Expressions ALGEBRA 2 LESSON 7-1 Quick Check Find all the real roots. Since 0.33 = 0.027, 0.3 is the cube root of 0.027. Since (–5)3 = –125, –5 is the cube root of –125. Since 54 = 625 and (–5)4 = 625, 5 and –5 are fourth roots of 625. There is no real number with a fourth power of –0.0016. 7-1

  5. a. –1000 Rewrite –1000 as the third power of a number. 3 Definition of nth root when n = 3, there is only one real cube root. = –10 3 = (–10)3 b. –81 Roots and Radical Expressions ALGEBRA 2 LESSON 7-1 Find each real-number root. There is no real number whose square is –81. Quick Check 7-1

  6. a. 9x10 9x10 =32(x5)2 = (3x5)2 = 3| x5| Absolute value symbols ensure that the root is positive when x is negative. 3 b.a3b3 3 (ab)3 = ab Roots and Radical Expressions ALGEBRA 2 LESSON 7-1 Simplify each radical expression. Absolute value symbols must not be used here. If a or b is negative, then the radicand is negative and the root must also be negative. 7-1

  7. c.x16y4 4 x16y4 = (x4)4(y)4 = x4 |y| 4 4 Roots and Radical Expressions ALGEBRA 2 LESSON 7-1 (continued) Absolute value symbols ensure that root is positive when y is negative. They are not needed for x because x4 is never negative. Quick Check 7-1

  8. < < < < < < < < < < – – – – – – – – – – 10 w 11 Write an inequality. d3 5 10 11 Substitute for w in terms of d. 50 d3 55 Multiply by 5. d3 5 50 d3 55 Take cube roots. 3 3 3 3.68 d 3.80 The diameters range from 3.68 in. to 3.80 in. Roots and Radical Expressions ALGEBRA 2 LESSON 7-1 A cheese manufacturer wants to ship cheese balls that weigh from 10 to 11 ounces in cartons that will have 3 layers of 3 cheese balls by 4 cheese balls. The weight of a cheese ball is related to its diameter by the formula w = , where d is the diameter in inches and w is the weight in ounces. What size cartons should be used? Assume whole-inch dimensions. Find the diameter of the cheese balls. 7-1

  9. Roots and Radical Expressions ALGEBRA 2 LESSON 7-1 (continued) The length of a row of 4 of the largest cheese balls is 4(3.80 in.) = 15.2 in. The length of a row of 3 of the largest cheese balls 3(3.80 in.) = 11.4 in. The manufacturer should order cartons that are 16 in. long by 12 in. wide by 12 in. high to accommodate three dozen of the largest cheese balls. Quick Check 7-1

  10. 3 4 3 1 25 1 216 Roots and Radical Expressions ALGEBRA 2 LESSON 7-1 1. Find all the real square roots of each number. a. 121 b. –49 c. 64 d. – 2. Find all the real cube roots of each number. a. –8000 b. 3. Find each real-number root. a. 0.49 b. 125 c. – 81 d. –625 4. Simplify each radical expression. a. –8x3b. 16y4c. 36x14 5. The formula for the volume of a cone with a base of radius r and height r is V = r 3. Find the radius to the nearest hundredth of a centimeter if the volume is 40 cm3. ± 11 none ± 8 none 1 6 –20 5 –9 0.7 none 6 | x7 | 4y2 –2x 1 3 about 3.37 cm 7-1

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