FACT SHEET: PROPERTIES OF ROOTS & RADICALS Definitions: ● “ b = n a ” means bn = a . We say “b is the nth root of a”. ● In the expression n a , n is the index, and a is the radicand. Even index: ● If radicand is negative, the number is not real. Ex.: − 4 and 6 − 7 are not real #’s. (Can you think of a real number n such that 푛2 = −4 ? No!) ● If the radicand involves an unknown, use absolute value bars in your answer if the answer calls for the principal root. Ex.: (x + 3)2 = x + 3 . ● To find the domain of a function, set the radicand greater than or equal to zero 4 and solve. Ex.: f (x) = 3 − x → Set 3 − x 0 & solve for x to find domain. Odd index: ● If the radicand is negative, the number is still real. Ex.: 5 − 32 = –2. [Since (−2)5 = (−2)(−2)(−2)(−2)(−2) = −32 ] ● If the radicand involves an unknown, absolute value bars are not necessary. Ex.: 7 x 7 = x , not |x| (if x is negative, we want it to remain negative in the answer). ● The domain of the function is simply , the set of real numbers. Ex.: g(x) = 5 3 − x has domain . Rational Exponents: 1 푚 푚 • 푎 ⁄푛 = 푛√푎 • 푎 ⁄푛 = (푛√푎) = 푛√푎푚 1 4 2 3 2 Ex.: 16 ⁄4 = √16 = 2 Ex.: 8 ⁄3 = (√8) = 22 = 4 , or 2 3 3 8 ⁄3 = √82 = √64 = 4 Multiplication and Division: ● n a n b = n ab [Note: Indexes must be alike! Multiply the radicands] n a a ● = n (푏 ≠ 0) [Note: Indexes must be alike! Divide the radicands] n b b 3 3 3 3 √98 98 Ex.: √5 ∙ √6 = √5 ∙ 6 = √30 Ex.: = √ = √49 = 7 √2 2 Counterexample: 3√10 ∙ 4√10 ≠ 12√10 Addition and Subtraction: !!!!!!!!!! WARNING: n a b n a n b !!!!!!!!! ● You may only add/subtract like radicals, which have same index and same radicand. Ex.: 5 3 xy + 7 3 xy = 12 3 xy (add/subtract coefficients, as with polynomials) 3 4 7 4 x − 4 y 4 x − y Counterex’s: x + x x ; Rationalizing Denominators: a ● To put into proper form, multiply the top and bottom by b : b 푎 푎 √푏 푎√푏 3 3 √5 3√5 = ∙ = Ex.: = ∙ = √푏 √푏 √푏 푏 √5 √5 √5 5 a ● To put into proper form, multiply top & bottom by the conjugate of b + c : b + c a a (b − c) = … and FOIL the denominators carefully. b + c (b + c) (b − c) 15 15 (4−√6) 15(4−√6) 15(4−√6) 3(4−√6) 12−3√6 Ex.: = ∙ = = = = 4+√6 4+√6 (4−√6) 16 − 4√6 + 4√6 − 6 10 2 2 3 √6 (or 6 − 2 ) Principle of Powers: ● If a = b, then an = bn . ● Steps for solving a radical equation: Ex.: √푥 − 4 + 7 = 10 – 7 – 7 1) Isolate the radical on one side of the equation. 1) √푥 − 4 = 3 2 2) Use the principle of powers to eliminate the 2) (√푥 − 4) = 32 th radical (if the radical is an n root, raise both sides of the equation to the nth power). → 푥 − 4 = 9 3) Solve the resulting equation. 3) + 4 + 4 4) Always check for extraneous (false) solutions! → 푥 = 13 4) Check: √13 − 4 + 7 = 10 ? In this case, the solution x = 13 is true. If it were false, √9 + 7 = 10 ? we would eliminate the answer and say “no solution”. 3 + 7 = 10 ✓ .