RADICALS & RATIONAL EXPONENTS

Transcription

1 c Gabriel Nagy RADICALS & RATIONAL EXPONENTS Facts about Power Equations Consider the power equation x N #, with N > integer and # any real number. Regarding the solvability of this equation, one has the following cases I. If N is odd, the equation has one real solution, no matter what the value of # is. II. If N is even, then (a) if # 0 : the equation has one real solution x 0; (b) if # > 0 : the equation has two real solutions; (c) if # < 0 : the equation has no real solutions. EXAMPLE : The table below contains several equations, their solutions, and the applicable cases Radical Notation Equation Solution(s) Case x 3 7 x 3 I x 3 x I x 0 x 0 II (a) x x, II (b) x 6 x, II (b) x no real sol. II (c) With N and # as above, the notation N # designates one special solution of the power equation x N #, namely: I. If N is odd, the unique solution is chosen

2 II. If N is even, the non-negative solution is chosen. This is applicable only when # 0. In the case when # < 0, the radical N # is undefined. Convention: When N, the radical # is simply denoted by # EXAMPLE (cont.) Based on the discussion on the equations in Example, we have: undefined. Properties of Radicals Undoing Formulas for Radicals.. ( N a) N a a if N is odd N. an a if N is even Arithmetic Formulas for Radicals.. N ab N a N b N a N a. b N b 3. N ap ( N a ) p (Assuming N a and N b are defined.) (UFR) (AFR) WARNING! What is incorrect about the equality a a? The above equality holds only when a 0. For instance, if we try a, we have a ( ) a. By (UFR) we know that a a. TIP: The above formulas can be used when we want to simplify N Expression, where Expression involves powers, products and quotients. We do so by

3 factoring N-powers out of Expression, then pulling N-powers outside N Expression, using (UFR) and (AFR). EXAMPLE : Suppose we want to simplify 6x y z 3. Using the equalities , x (x ) x, and y (y ), we see that we ( 6x y ) can factor the square under the radical, then we pull it out: z (36x 6x y y ) ( ) (6x 6x y ) ( ) 6x 6x y 6x z 3 z z z z z z. NOTE: Since N is even, we pulled out the square using the absolute value. If all variables x, y, z are assumed to be positive, the above simplification can be continued (with last equality optional) as: 6x y 6x y 6x z 3 z z 6x y 6 x z. z Fractions with radicals Suppose we have a fraction, such as the one in Examples 3 and below, and we are asked to transform it (by multiplying both sides by the same Expression) into an equivalent one Numerator Denominator (Numerator) (Expression) (Denominator) (Expression), so that the new fraction is nicer. The specification of what nicer means is often formulated as a request: rationalize the denominator, or rationalize the numerator. This means that we look for and expression, such that one of the products (Numerator) (Expression) or (Denominator) (Expression) is without radicals. 3

4 EXAMPLE 3: If we want to rationalize the denominator in 3, the expression is pretty obvious: (using 3 3 3): TIPS: For more complicated Numerators or Denominators, try one of these identities (based on (a + b)(a b) a b ): Rationalizing tricks.. ( A + B)( A B) A B. ( A + C)( A C) A C (RT) EXAMPLE : To rationalize the denominator in we multiply both sides by ( 3 + ). Note that in the new numerator we have a square, for which we employ the Square of a Sum Formula ((a+b) a + ab + b ): 3 + ( 3 + ) ( 3 + ) 3 ( 3 + ) ( 3 ) ( 3) ( ) Fractional Powers (Rational Exponents) + 6. Suppose M N is a rational number (with the fraction simplified). The M N powers are computed using one of the equalities: th a M N ( N a ) M N a M, provided N a is defined In particular, for an integer N >, the th N powers are computed by a N N a, provided N a is defined

5 EXAMPLE : To compute 3 3 we use the definition with M 3 and N, combined with (UFR) and the equality 3 : 3 3 ( 3) 3 ( ) EXAMPLE 6: To convert xy 3 to radical notation, we use the second equality in the definition (for x 3 we use M 3, N ; for y we use M, N ), then we simplify (last equality optional) using (AFR): x 3 y x 3 y x 3 y x 3 y. Arithemtic of Fractional Powers FACT: The formulas first introduced in R, collected in the table below, also work if the epxonents m and m are rational. Arithmetic Formulas for Powers.. a n a n. a m a n a m+n 3. am a n am n. (a m ) n a mn. (ab) m a m b ( m a m a 6. m b m (Provided all powers are defined.) (AFP) EXAMPLE 7: To simplify a 3 a we add the exponents using (AFP): a 3 a a 3 + a 7 6. (How did we get 7 6? When adding 3 and we converted them to fractions with denominator 6, that is: ) EXAMPLE 8: To convert 6x w 3 to rational exponents, we replace directly the operation with the th power, then (note that 6 ) we use (AFP): 6x w 3 ( 6x w 3 ) ( ) (x ) (w 3 ) x w 3 x w 3.

6 EXAMPLE 9: To convert x 3 y 6 to radical notation, we first simplfy (see x 3y also R for the technique of handling products and quotients of powers) the expression using (AFP), then convert the factors to radicals : [ ] [ ] x 3 y 6 x 3 y [ 6 x x 3y x y 3 ( 3) ] [ ] [ ] y 6 [x] y 3 x 3 y. (Make sure you understand the calculations of 3 ( 3) and 6 3.) 6