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6.2 Multiplying and Dividing Radical Expressions I. Multiplying Radical Expressions Property: If n√a and n√b are real numbers, then n√a (n√b) = n√ab Example 1: Multiplying Radicals (Simplify if Possible) 1a) √2 (√8) = √ 2 * 8 = √16 = 4 1b) ³√-5(³√25) = ³√-5 * ³√25 = ³√-125 = -5 1c) √-2 (√8) = the property for multiplying radicals does not apply. √-2 is not a real number. 1d) √3(√12) 1e) ³√3(³√-9) 1f) 4√4(4√-4) Example 2: Simplifying Radical Expressions Assume that all variables are positive. Then absolute value symbols are never needed in the simplified expressions. 2a) √72x³ Step 1: Factor into perfect square (or to nth index) √72x³ = √6² * 2 * x² * x Step 2: ^n√a (^n√b) = ^n√ab √72x³ = √6² * x² (√2 * x) Step 3: Simplify 6x√2x 2b) ³√80n5 2c) √50x4 2d) ³√x³ (Assume that x is positive for both ex.’s) Example 3: Multiplying Radical Expressions Multiply and Simplify ³√54x²y³ ( ³√5x³y4). Assume that all variables are positive. Step 1: Multiply all coefficients and variable w/ exponents. ³√54x²y³ * 5x³y4 = ³√270x5y7 Step 2: Factor it down by factor by Prime Factor Tree or Perfect Cubes 3b) 3√7x³ * 2√21x³y² 3c) ³√25xy8* ³√5x4 y³ II. Dividing Radical Expression If n√a and n√b are real numbers and b ≠ 0, then n√a/n√b = n√a/b Example 4: Dividing Radicals Divide and Simplify. Assume that all variables are positive. 4a) ³√32/³√-4 = ³√-8 = -2 4b) Please remember when dividing exponents, you are subtracting them! ³√162x5 ³√3x² = √54x³ (simplify it down and get your answer) 4c) 4√1025x15 4 √4x Rationalize the Denominator: In an expression, rewrite it so there are no radicals in any denominators and no denominators in any radical; rationalizing the denominator of numerical expression makes it easier to calculate its decimal approximation. Example 5: Rationalize the denominator of each expression. *Assume that all variables are positive. 5a) √2/√3 Step 1: Multiply the numerator and denominator by √3 (which is the denominator). The denominator will become a whole number. √2 (√3) √6 √3 (√3) = 3 5b) √x³ √5xy Cubic Root or Higher: If the denominator is not a perfect cube or nth root of the number, you must rewrite it into one. Whatever was used to convert into a perfect cube or nth root must be multiplied to the numerator as well. 5c) ³√2 ³√4 3x ³√6x