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7.1 and 7.2 Roots and Radical Expressions and Multiplying and Dividing Radical Expressions 1 Roots and Radical Expressions • Since 52 = 25, 5 is a square root of 25. • Since (-5)2 = 25, -5 is a square root of 25. • Since (5)3 = 125, 5 is a cube root of 125. • Since (-5)2 = -125, -5 is a cube root of -125. • Since (5)4 = 625, 5 is a fourth root of 625. • Since (-5)4 = 625, -5 is a fourth root of 625. • Since (5)5 = 3,125, 5 is a fifth root of 3,125. And the pattern continues……. 2 Roots and Radical Expressions • This pattern leads to the definition of the nth root. • For any real numbers a and b, and any positive integer n, if an = b, then a is an nth root of b. • Since 24 = 16 and (-2)4 = 16, both 2 and –2 are fourth roots of 16. •Since there is no real number x such that x4 = -16, -16 has no real fourth root. •Since –5 is the only real number whose cube is –125, -5 is the only real root of –125. 3 Roots and Radical Expressions Type of Number Number of Real Number of Real nth Roots when nth Roots when n is Even n is Odd Positive 2 1 0 1 1 Negative None 1 4 Finding All Real Roots • Find all real roots. • The cube roots of 0.027, -125, 1/64 • The fourth roots of 625, -0.0016, 81/625 • The fifth roots of 0, -1, 32 • The square roots of 0.0001, -1, and 36/121 5 Radicals • A radical sign is used to indicate a root. • The number under the radical sign is called the radicand. • The index gives you the degree of the root. • When a number has two real roots, the positive root is called the principal root and the radicand sign indicates the principal root. 6 Radicals • Find each real – number root. 27 3 4 81 49 16 4 7 Radicals • Find the value of the expression of x = 5 and x = -5 x 2 • For any negative real number a, n a n a when n is even. 8 Radicals • Simplify each radical expression. 4x 3 6 a3b6 10 9x 3 a3b3 9 Radicals • Simplify each radical expression. 16 4 x y 4 x4 y8 4 4x2 y4 3 27c 6 4 x 8 y12 10 Radicals are the inverse of exponents Exponents: Radicals: 25 5 5 25 2 5 125 3 3 125 5 5 625 4 625 5 5 3125 5 3125 5 4 5 11 Simplify the Radicals 125 5 5 3 250 5 2 3 3 3 2000 3 10 3 2 30 3 2 5 640 5 2 20 12 Rules for Simplifying Radicals • Square roots can simplify if there are sets of two duplicate factors. • Cube roots can simplify if there are sets of three duplicate factors. • Fourth roots can simplify if there are sets of four duplicate factors. • Fifth roots can simplify if there are sets of five duplicate factors. • And so on and so forth…. 13 Simplify the Radicals 7 7 7 3 8 2 8 68 48 14 28 14 14 2 14 2 15 3 12 5 3 3 4 3 32 5 3 6 15 14 Simplify the Radicals 8 2 4 4 8 2 2 3 4 3 3 2 3 12 3 3 4 3 45 3 3 2 9 3 3 3 2 3 2 2 8 3 323 2 3 5 3 3 3 4 3 5 3 3 27 3 63 10 15 Simplify the Radicals 16 2 4 4 4 4 4 5 4 9 4 4 4 18 4 10 4 9 4 4 4 9 3 81 4 4 2 4 2 4 2 16 4 324 5 64 5 16 Simplify the Radicals 2 128 7 2 2 256 x x x 7 7 256x y z 2xy 2xz 8 7 2 7 2 8 3 256x 8 y 7z 2 4x 2 y 2 3 4x 2 yz 2 7 2 2 2 256 x x x x y y y y 3 8 7 3 2 3 3 3 2 3 17 Simplify the Radicals 5 243x18 y 6z12 3x 3 yz 2 5 x 3 yz 2 3 256x 8 y 7z 2 4x 2 y 2 3 4x 2 yz 2 18 Simplify the Radicals 9 3 6 4 | x y | z 2 32x y z 18 15 6 12 10x 80 y 34 z18 x 5 y 2 z15 10 x 5 y 2 z 3 19 Simplify the Radicals 32x12 y 3 z12 5 3 8 8x y z 4 x 7z 4 2 | x 3 | z 2 x Most of the time, it is easier to divide first, then simplify later. 5 128x15 y13z 8 5 2 6 2xy z 5 64 x14 y 11 z 2 2 x 2 y 2 5 2 x 4 yz 2 20 Simplify the Radicals 6 64 x 29 y 31z10 6 15 13 16x y 6 2 3 2 4 6 x | y | z 4 x z 4x y z 25 x104 y 30z 22 14 18 10 25 x105 y 32z 28 25 2 6 xy z x 4 y 6 z 4 25 x 4 z 22 21 Rationalizing the Denominator It is considered bad form to have a radical in the denominator of an expression. It is necessary to do some algebra so that there is no longer a radical in the denominator. 8 2 This should not be here. 22 Rationalize the Denominator To rationalize the denominator, you usually have to multiply by a fraction that is equal to one that also contains numbers that allow the offending radical to be removed. 8 2 8 2 4 2 Multiply by: 2 2 8 2 2 2 4 2 23 Rationalize the Denominator 7 7 5 2 2 5 5 5 2 35 25 Multiply by: 5 5 2 35 5 24 Rationalize the Denominator 3 x 2x 10 xy 2 5y 5y 5y Divide first Multiply by: 5y 5y 2 5x y 25 y 2 2 5x y | x | 5 y 5| y | 5| y | 25 Rationalize the Denominator 3 3 3 3 4 6x 18 x 2 27 x 3 3 3 2 3x 3 3 3 18 x 3x (3x) 2 Divide first (3x) 2 Now multiply to rationalize the denominator. 2 26 Rationalize the Denominator 3 3 3 3 12 5x 300 x 2 125 x 3 3 (5 x) 2 3 (5 x) 2 3 300 x 5x Multiply to rationalize the denominator. 2 27 Rationalize the Denominator 3 3 3 3 10 3x 3 3 2 90 x 27 x 3 9x 9x 3 Multiply to rationalize the denominator. 90 x 3x 28