Algebra 2 Lesson 7-1 (Page 363) ALGEBRA 2 LESSON 7-1 Roots and Radical Expressions 7-1

To simplify nth roots. ALGEBRA 2 LESSON 7-1 Roots and Radical Expressions 7-1

For any real numbers a and b, and any positive integer n, if Then a is an nth root of b. ALGEBRA 2 LESSON 7-1 Roots and Radical Expressions 7-1

ALGEBRA 2 LESSON 7-1 Roots and Radical Expressions 7-1 Type of Number Number of Real nth Roots when n is even. Number of Real nth Roots when n is odd. Positive21 Zero11 NegativeNone1

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

Roots and Radical Expressions ALGEBRA 2 LESSON 7-1 Find all the real roots. 7-1 a.Find all the real fifth roots of 0, -1, and 32. a. 0; no real fifth root; 2 b. 0; –1; 2 and –2 c. 0; –1; 2 d. 0; –1; no real fifth root

Roots and Radical Expressions ALGEBRA 2 LESSON 7-1 Find all the real roots. 7-1 b.Find all the real square roots of , -1, and a and –0.01; no real square root; and b. 0.01; –1; c. 0.01; no real square root; d. 0.01; no real square root; and

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

Roots and Radical Expressions ALGEBRA 2 LESSON 7-1 Find each real-number root. 7-1 a. no real-number root b. –3 c. 3 d. 9 a. no real-number root b. 7 c d. –7

ALGEBRA 2 LESSON 7-1 Roots and Radical Expressions 7-1

When a number has two real roots, The positive root is called the ALGEBRA 2 LESSON 7-1 Roots and Radical Expressions 7-1

For any negative real number a, When n is even. ALGEBRA 2 LESSON 7-1 Roots and Radical Expressions 7-1

3 b.a 3 b 3 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. a.9x 10 Absolute value symbols ensure that the root is positive when x is negative. 9x 10 =3 2 (x 5 ) 2 =(3x 5 ) 2 = 3| x 5 | (ab) 3 = ab 3 7-1

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 x 2 is never negative. c.x 16 y x 16 y 14 =(x 4 ) 4 (y) 4 = x 4 |y| 7-1

Roots and Radical Expressions ALGEBRA 2 LESSON a. –2 | x | y 2 b. 2 | x | y 2 c. | x | y 2 d. 2xy 2 a. 3c 2 b. 3c 3 c. 3| c 3 | d. –3c 2 a. x 2 | y 3 | b. x 2 y 8 c. x 2 y 3 d. x 2 | y 8 |