Integer Exponents. Examples: 5 3 = = 125, Powers You Should Know

Transcription

1 Algebra of Exponents Mastery of the laws of exponents is essential to succee in Calculus. We begin with the simplest case: 200 Doug MacLean Integer Exponents Suppose n is a positive integer. Then a n is simply means n copies of a multiplie together, i.e., a n a a a... a a a }{{} n copies of a Examples: , ( ) It is then clear that the law a m+n a m a n must hol when m an n are positive integers, because a m+n a a... a a a }{{} m + n copies of a an a m a n } a a... {{ a a a} } a a... {{ a a a} m copies of a n copies of a mean exactly the same thing. Powers You Shoul Know From your knowlege of the multiplication tables from to 0 you automatically know ten secon powers, or squares: 2, 2 2 4, 3 2 9, 4 2 6, , , , , 9 2 8, an In aition, you also know four thir powers, or cubes: 3, 2 3 8, , , an three fourth powers: 4, 2 4 6,

2 The higher powers of 2 shoul be known at least up to the tenth power: 200 Doug MacLean , , , , , an The powers of 0 are of course really easy: 0 n is just a followe by n 0 s. Any number raise to the zero-th power is efine to be, i.e., a 0 Examples: 4 0, π 0, ( ) 0 4 Note that the law a m+n a m a n is still true when either m or n, or both, equal 0. Negative Integer Powers We efine a a an a n a n so that the law a m+n a m a n is still true when either m or n or both are integers less than 0. 2

3 In aition, we have another useful law for quotients of powers of the same number a: a m a n am n 200 Doug MacLean which is true for all possible integers m an n, be they positive or negative. Roots The n-th root of a positive number a is that positive real number a n a, i.e., (also written n a) which, when raise to the n-th power, returns ( a n) n a In particular: ( ) a 2 ( 2 a, a 3) 3 a. As an example, 8 3 is that positive number which, when raise to the thir power, gives 8. Since 2 3 8, then 2 is the number we are looking for, i.e., We must be very careful to use precise language. We say that 2 is the fourth root of 6, an we say that 2 is a fourth root of 6. Fractional Powers Suppose a is any positive real number, an r is any rational number, i.e., r m, the quotient of two integers m an n, where n 0. n We efine a r a m n ( a n) m, an call it a raise to the r -th power, or a to the m over n. 3

4 Examples 200 Doug MacLean (27 3 ) (5 2 ) ( ) 3 ( Even with these fractional exponents, we still have the law: ) 4 3 [ ] a r ( ) r a a r Examples ( ) (3) ( ) 5 3 ( ) The Laws of Exponent Algebra Aition an Subtraction of Exponents 4

5 a r +s a r a s an a r s ar a s a s r ( ) Examples: 5 +s 5 (5 s ) 5 2 s s 5 s Multiplication an Division of Exponents a rs (a r ) s (a s ) r an a r s (a s ) r (a r ) s Examples: 2 2r ( 2 2) r 4 r 8 r 3 (8 3 Multiplication an Division of Bases ) r 2 r 2 r (ab) r a r b r an ( ) a r b a r b r. Memorizing these rules is not ifficult if you keep in min the integer exponent case. For example, suppose you nee to simplify ( 3 r + ) r but cannot recall if the rule for (a r ) s is a rs or a r +s. Just check the formulas in a simple case, say with a 2, r, an s 2: (a r ) s ( 2 ) , a rs , which are equal, but a r +s This shoul pretty quickly help you remember (a r ) s a rs. Thus ( ) 3 r + r 3 (r +)(r ) 3 r 2. 5

6 2. Although a r has only been efine for a a positive real number, in some cases a r oes make sense when a is negative. For example, ( 2) 3 8, an thus ( 8) 3 2. On the other han, ( 8) 2 is not efine (unless we use complex numbers). In general, a m n will be efine for negative a if n is an o integer, but not if n is an even integer. For these reasons the rules are not always vali for negative bases an must be hanle with care! As an example, when a is negative (say a 2) it is NOT true that ( a 2) 2 a; instea ( a 2) 2 a. 200 Doug MacLean 3. A major goal of calculus is to efine a r for r any real number, not just for r a rational number. It is then amazing but true that the rules remain vali. ( 4. The rule a r a in the following simplification: ) r a r is quite useful in allowing us to eliminate negative exponents in fractions. Observe its use (x + 2) 3 (x 2) 3 (x 2)3 (x 2 4) 2 (x 2 4) 2 (x + 2) 3 where the terms with negative exponents have switche between the numerator an enominator: (x 2)3 [(x 2)(x + 2)] 2 (x 2)3 (x 2) 2 (x + 2) 2 (x + 2) 3 (x + 2) 3 (x 2)5 (x + 2) 2 (x 2)5 (x + 2) 3 x + 2 Examples: (4a) a 2 2 a, ( a 2 8 ) 2 3 ( a 2 ) a Non-Formulas for the Aition an Subtraction of Bases 6

7 There are (a + b) r? an (a b) r? NO simple formulas in these cases. For instance, (a + b) r oes NOT equal a r + b r. This is a very common an fatal error, so be very, very careful!! n-th Roots the notation An important special case of exponents occurs when r n for n a positive integer. In that case we often use a n n a an refer to n a as the n-th root of a. (when n 2 we simply write a in place of 2 a.) From its efinition we have the basic formulas: ( n a ) n n a n a for any positive real number a. We also have n ab n a n b an n a b n a n b Again, we emphasize that there are NO simple formulas for n a + b or n a b Don t ream anything up for these expressions!! Examples Example Simplify x(x + ) 5 (x + ) 6 5 x 2 7

8 Using , an x2 (x + )(x ), we rewrite the expression as x(x + ) 5 (x + ) + 5 (x + )(x ) (x + ) 5 [x (x + )] (x + )(x ) x(x + ) 5 (x + )(x + ) 5 (x + )(x ) ( ) (x + ) 5 (x + ) (x ) (x + ) 4 5 (x ) Example 2 Simplify 4 (x 2) 5 + (x 2) 5 x This is a fairly common type of expression. As in the previous example we have the same base raise to ifferent fractional powers. In general the way to procee is to factor out the lowest fractional power in this case (x 2) 5 from both terms in the numerator, by using the fact that (x 2) 5 (x 2) 5 : (x 2) (x 2) 5 (x 2) 5 + (x 2) 5 x x [ ] (x 2) 5 + (x 2) 5 (x 2) 5 (x 2) 4 5 x ] (x 2) 5 + (x 2) 5 [(x 2) 4 5 (x 2) 5 x (x 2) 5 + (x 2) 5 (x 2) x (x 2) 5 + (x 2) 5 (x 2) x (x 2) 5 [ + (x 2)] x (x 2) 5 (x ) x (x 2) 5 5 x 2 Example 3 Simplify 3 8a 6 (x h) 4 + 2a 2 h 3 x h the first term: The trick in this type of an expression is to pull as much out of the cube root as possible. We start by examining 8

9 3 8a 6 (x h) a (x 6 3 h) (a ) 3 3 (x h) 3 (x h) 2a 2 (x h) 3 x h Thus our full expression becomes 2a 2 (x h) 3 x h + 2a 2 h 3 x h 2a 2 3 x h(x h + h) 2a 2 x 3 x h Algebraic Conjugates The terms () c an c + are calle algebraic conjugates of each other, as are pairs of terms like (2) a b an a + b an (3) a b an a + b. The ifference of squares law gives us their proucts: () (2) ( c )( c ) + ( c ) ( 2 2 ) c ( )( ) ( 2 a b a + b a 2 b) a 2 b (3) ( a b )( a + b ) ( a ) 2 b 2 a b 2 9

10 Example 4 Rewrite c + as a fraction with no roots in the enominator. 200 Doug MacLean Multiply by in the form of the conjugate of the enominator over itself: c + ( c c ) c ( c + )( c ) c c. Example 5 Rewrite c as a fraction with no roots in the enominator. Multiply by in the form of the conjugate of the enominator over itself: c ( c + c + ) c + ( c )( c + ) c + c. Example 6 Simplify x + x. Multiply by in the form of the conjugate of the enominator over itself: x + x ( x + + x x + + x ) x + + x ( x + x)( x + + x) x + + x x + x x + + x x + + x. Example 7 Remove all square roots from the enominator of We must rationalize the enominator. h x + h x 0

11 h x + h x ( ) h x + h + x x + h x x + h + x ( ) h x + h + x ( )( ) x + h x x + h + x ( ) h x + h + x (x + h x) ( h ) x + h + x h x + h + x Example 8 Is it always true that a 2 a? Answer: NO. Let a. Then a 2 ( ) 2 a 2 is calle the absolute value of a an is never negative. It is written a. Many calculus errors are mae when an expression of the form a 2 is incorrectly replace by a.

12 Example 9 Determine all x values for which (x ) Doug MacLean We know our equation is true if (x ) 2. Since the only numbers whose square is are an, we either have x orx so either x 0orx 2. 2