Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Important Radical Information By definition, an nth root of a number is this: The number c is an nth root of a if c n a . EX: The square roots of 81 are 9 and -9, because 9 2 81 and (9) 2 81 The index of a radical symbol n EX: 3 indicates what root is being found. n x means the “cube root” of x. Similarly, 7 x means the “7th root” of x. ODD ROOTS: Every real number has just one real odd root (index is odd): odd roots of positive numbers are positive and odd roots of negative numbers are negative. EX: 5 32 = 2 and 5 32 2 . It’s not a problem to have a negative radicand with an odd root. EVEN ROOTS: Every positive real number has two real even roots (index is even). Negative numbers do not have real nth roots when n is even. EX: The 4th roots of 16 are 2 and -2. Negative 16 (-16) does NOT have a 4th root, since no real number multiplied to itself 4 times can be negative. The chart below summarizes the rules for radicands, depending on the index, and how to find the domain of a radical function. REMEMBER: the domain rules only apply if the function does not contain the radical in any denominator. Index Radicand possibilities To find domain: Even Must be non-negative 0 . Set radicand 0 and solve for the variable The even root of a negative number does not exist Odd Can be any real number Domain can be any real number Any positive number has two square roots. However, the symbol indicates the “principal,” or positive square root of a number only. The secondary (negative) root is indicated by the symbol . The same principal applies for any even index. EX: Even though 16 had two square roots (4 and -4), 16 ONLY equals 4, NOT -4. If I want to refer to the negative root (-4), I would need to say 16 4 SIMPLIFYING RADICALS involves removing perfect nth power factors from the radicand. The best way to do this is to factor the radicand in such a way that all factors in the radicand are written as a perfect nth power (n or a multiple of n), or as something to a power less than the index n. SIMPLIFY 3 135 x 5 y 10 Factor 135 into primes 3 33 5 x 5 y 13 EXAMPLE: Rewrite variable factors either with exponents that are multiples of the index (3) or with exponents less than the index (3) 3 33 5 x 3 x 2 y 12 y 3 33 x 3 y 12 3 5 x 2 y Put all perfect cube factors under one radical, and all other factors under a separate radical 3 xy 4 3 5 x 2 y Simplify the radical containing perfect cubes. PRODUCT RULE OF RADICALS: n ab n a n b . This rule is used for simplifying (as shown above), as well as for multiplying EX: 3x 12 x 2 36 x 3 6 2 x 2 x 6 2 x 2 x 6 x x QUOTIENT RULE OF RADICALS: EX: 2 150 xy 3x 2 n a b n a n b 150 xy 2 50 y 2 25 2 y 2 25 2 y 2 5 2 y 10 2 y 3x ADDING AND SUBTRACTING: Only “like radical terms” (ones in which the radicand and the index are identical) can be added or subtracted. This concept is very similar to the idea of adding and subtracting like terms. When added, the radical will be the SAME as it was in the like radical terms. Be sure to simplify radical expressions before attempting to add or subtract. Remember about invisible 1’s in front of radicals if no other “coefficient” is showing. 9x 9 x 1 EX: 9( x 1) x 1 3 x 1 x 1 Invisible “1” in front of second radical. 4 x 1 PYTHAGOREAN THEOREM: For any right triangle with legs a and b and hypotenuse c, a 2 b2 c2 c a b