The radicand cannot be a fraction or decimal the

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Unformatted text preview: ecimal, the denominator cannot contain a radical. examples: Simplify 30 6 = 30 6 = 5 18 75 6 25 = b) = 6 25 = a) 6 5 or 16 5 When the denominator of a fraction is an irrational number, the denominator is a nonterminating and non-repeating number. It is impossible to divide by such a decimal number. The process or rewriting a fraction so that the denominator is not irrational is called rationalizing the denominator. examples: Simplify a) 5 7 = = = 5 7 (5)(7) (7)(7) 35 7 7 7 = b) 15 20 = 15 25 15 = 2 5 15 5 = 2 (5)(5) 1 35 7 15 5 10 35 = 2 5 5 = exercises: Simplify 56 a) 2 11 5 66 22 b) 57 2 90 70 12 c) 5 34 = 35 2 53 2 2 Unit 4: Day 3 notes - Dividing Radical Expressions Page 2 of 2 CONJUGATES The conjugate of a binomial is another binomial with the same first term, but the second term has the opposite coefficient (opposite sign); a + b and a − b . The product of conjugates (a + b)(a −b) is a2 − b2 . example: Multiply ( 7 3 − 2 5 ) to its conjugate. (7 3 − 2 5 ) (7 3 + 2 5 ) (7 3) 2 (2 − − 127 147 Answer: 5) 2 20 Note that the product of binomial radical conjugates is a rational number. DIVIDING RADICAL EXPRESSIONS Consider the division 6 ÷ (1 + 5 ) ; this division results in the quotient 6 1+ 5 This fraction is not in simplest form because there is a radical in the denominator. example: Simplify 6 1+ 6 1+ 5 6 (1 − o Use the conjugate to simplify (1 + 5 5) 5 ) (1 − 6 (1 − 5) 5) (1)2 − ( 5 ) 2 6 (1 − 5 ) −4 − 3 (1 − 5 ) 2 Answer: exercise: Simplify −3+3 5 2 2 3 −3 2 2 3 +3 2 = − 33 + 5 22 [Answer: −5 + 2 6 PRE-CALCULUS 11 Unit 4 – Day 4: RADICAL EQUATIONS (Part 1) RADICAL EQUATIONS A radical equation is an equation having radicals with the variable in the radicand. The solution of any equation are all values of the variable that satisfies the equation. 9− x + 5 = 2 is an example of a radical equation. Since there is a square root of a variable expression, there may be restrictions from the radicand: x + 5 ≥ 0 ∴ x ≥ −5 This equation can be solved graphically: • Rewrite the equation so that one side is 0; graph y = (the non-zero side); the roots are the x-intercepts. • Graph y = (the left side) and y = (the right side); the roots ar...
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