Chapter 5 Radical Expressions and Equations

5.1 Working With Radicals The √ symbol is called the radical sign and the number inside the radical sign is called the radicand. Together they form a radical. Numbers like √2 and √5 are called radicals. Radicals in Simplest Form: A radical can be in simplest form if the radicand doesn’t contain a fraction or any factor which can be removed. Ex. √18 is not in simplest form because 18 has a square factor of 9, which can be removed. √18 is equivalent to the simplified form, 2√3. Adding and Subtracting Radicals: When adding or subtracting, combine coefficients of like radicals. In general, mr√a + nr√a = (m + n)r√a .

Example 1: Convert the mixed radicals into entire radical form. a) 7√2 b) a4√a a) 7√2 = √72(√2) b) a4√a = √(a4)2(√a) = √72(2) = √(a4)2(a) = √49(2) = √(a)8(a) = √98 = √a9 Example 2: Convert the entire radicals to a mixed radical in simplest form. a) √ 200 b) √ 48y5 a) √ 200 = √ 100 (2) b) √ 48y5 = √ 16(3)(y4)(y) = √ 100 (√ 2) = 4y2√ 3y = 10 √ 2

5.2 Multiplying and Dividing Radical Expressions When multiplying radicals, multiply the coefficients and multiply the radicands. But, you can only multiply radicals if they have the same index. Ex. 5√ 3 (√ 6) = (5 x 1)√ 3 x 6 = 5 √ 18 = 5 √ (9)(2) (5)(3)√ 2 15√ 2 Rationalizing Denominators To simplify an expression that has a radical in the denominator, you need to rationalize the denominator. If an expression has a monomial square-root denominator, multiply the numerator by the radical term from the denominator. If an expression has a binomial denominator that contains a square root, multiply the numerator and denominator by a conjugate of the denominator.

Example 1: Rationalize 5 Example 2: 2√ 3 5 = 5 (√ 3) 2√ 3 2√ 3(√ 3) 5 = 5 (√ 3) 2√ 3 2√ 3(√ 3) = 5√ 3 5√ 3 2√ 3 (√3) 6 Example 2: Rationalize the denominator 5√ 3 4 - √ 6 = ( 5√ 3 )( 4 + √ 6) ( 4 - √6)(4 + √ 6) = 20√3 + 5√18 = 20√3 + 15√2 42 - (√6)2 10 = 20√3 + 5√(9)(2) = 4√3 + 3√2 16 - 6 2

5.3 Radical Equations A radical equation is an equation with radicals that have variables in the radicands.The algebraic solution is based on the fact that if two real numbers are equal, then their squares are also equal. Solving Radical Equations isolate the radical on one side of the equation square each side, then solve the equation that results identify the extraneous roots and reject them Example: Solve the following radical equations. √ x + 2 - 5 = 0 b) √ x - 2 + 5 = 0 (√ x + 2)2 = (5)2 (√ x - 2)2 = (-5)2 x + 2 = 25 x - 2 = 25 x = 23 x = 27 Check: √(23) + 2 - 5 = 0 √(27) - 2 + 5 = 0 √ 25 - 5 = 0 5 + 5 = 0 5 - 5 = 0 10 ≠ 0 0 = 0 ✓