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P.3 Radicals and Rational Exponents Q: What is a radical? What is a rational number? A: A “radical” involves a root symbol, whereas a “rational number” involves a fraction. Definition of the Principal Square Root • If a is a nonnegative real number, the nonnegative number b such that b2 = a, denoted by b = a, is the principal square root of a. That is: 4=2 (since 2 squared = 4), not –2 (even though (-2) squared also = 4). Square Roots of Perfect Squares 2 a a Ex: Simplify (-3)2 Ex: Simplify x2 Ex: Simplify -32 Ans: 3, |x|, and “not a real number” or 3i The Product Rule for Square Roots • If a and b represent nonnegative real number, then ab a b and a b ab • The square root of a product is the product of the square roots. Ex: Compare and draw conclusions: 9+16 vs. 9*16 • Ex: Simplify a. 500 b. 6x3x c. 108x6y11 Ans: a) 10 5 b) 3x 2 c) 6 x3 y 5 3 y The Quotient Rule for Square Roots • If a and b represent nonnegative real numbers and b does not equal 0, then a a b b and a b a . b • The square root of the quotient is the quotient of the square roots. Ex: Simplify 100 9 (ans: 10/3) Example We can only add radical expressions if they contain “like terms”: The same number must be under the radical sign (the radicand), and it must have the same index. Then just like ordinary “like terms” we add the COEFFICIENTS and KEEP THE “LIKE” parts the SAME. Ex: Perform the indicated operation: 4 3 32 3 Ans: (4 1 2) 3 3 3 Ex: Perform the indicated operation: 724 + 26 = Ans: 16 6 Definition of the Principal nth Root of a Real Number n a b n means that b a • If n, the index is: even, and a is nonnegative (a > 0) then b is also nonnegative (b > 0) Ex: 4 625 5 odd, a and b can be any real numbers with the same sign (+ or -) Ex: 3 125 5 Q: What would we write if n is even and a is negative? (Ans: “not a real number”.) Finding the nth Roots of Perfect nth Powers If n is odd, n an a If n is even a a. n n It is only “necessary” to use the absolute value symbol if you are finding the even root of a variable (unknown). Ex: Simplify each of the following: ³(-2)3 Ans: -2 2 (-2)2 3-8x7y11 2x y 2 2x 24 2 416x8y3 5 -x10 32 x 2 33 y 3 xy 2 The Product and Quotient Rules for nth Roots • For all real numbers, where the indicated roots represent real numbers, n a b ab and n n n n a n a , b0 b b Q: Do you remember for what operation(s) you may NOT separate ( or reverse to put together) the numbers? (A: sum or difference.) Definition of Rational Exponents a1 / n n a. Furthermore, 1 1 1/ n a 1/ n n , a 0 a a The denominator of the rational exponent becomes the INDEX of the radical expression. Ex: Simplify the following: 4½ Ans: 2 (-8)(2/3) 4 5 x3 y 2 3 2 y or 5 x3 y 2 (2 y )(1/ 3) (250x9y7)1/3 25x (125x6)2/3 4 Definition of Rational Exponents a m/ n m n m ( a) a . n • The exponent m/n consists of two parts: the denominator n is the root and the numerator m is the exponent. Furthermore, a m/n 1 a m/n . Example: Ans: 1 8 2x Simplify 2(-8x12)-(2/3) Rationalizing the Denominator ONE TERM in the denominator: simplify, then multiply by whatever is needed to make a perfect root (ONE TERM). Ex: 20 20 15 20 15 4 15 15 3 15 15 15 TWO TERMS in the denominator (one is a square root): simplify, then multiply by the conjugate (TWO TERMS). Ex: 2 2 2 4 2 2 8 2 16 24 24 2 4 2 2 8 2 4 4 2 or 14 7 7 Simplified form for Radical Expressions: •NO radical sign in the denominator •NO fractions under the radical sign •NO exponents greater than the index under the radical sign •The index is reduced as low as possible