advertisement

Warm up • 1. Change into Scientific Notation • 3,670,900,000 • Use 3 significant figures • 2. Compute: (6 x 102) x (4 x 10-5) • 3. Compute: (2 x 10 7) / (8 x 103) Lesson11-1 Rational Exponents Objective: To use the properties of exponents To evaluate and simplify expressions containing rational exponents To solve equations containing rational exponents Nth Roots a is the square root of b if a2 = b a is the cube root of b if a3 = b Therefore in general we can say: a is an nth root of b if an = b Exponents n n=1: b =b n>1: b =b b b ... b n 0 n=0: b =1 1 If b 0, then b = n . b -n Properties of Exponents m and n are positive integers a and b are real numbers Property Definition Example Product aman=am+n (am)n=amn (163)(167)=1610 (93)2=96 Power of a power Power of a quotient (a/b)m=am/bm (b≠0) (3/4)5=35/45 Power of a product (ab)m=ambm (5x)3=53x3 quotient am/an=am-n 156/152=154 (a≠0) Example Evaluate: 3 3 6 3 4 7 1 3 4 4 7 6 3 4 3 3 5 Example Simplify: s t 4 7 3 x2 y5 x 2 4 12 21 s t x2 y5 8 x y5 x6 Nth Root • If b > 0 then a and –a are square roots of b • Ex: 4 = √16 and -4 = √16 • If b < 0 then there are not real number square roots. • Also b1/n is an nth root of b. • 1441/2 is another way of showing √144 • ( =12) Principal nth Root • If n is even and b is positive, there are two numbers that are nth roots of b. • Ex: 361/2 = 6 and -6 so if n is even (in this case 2) and b is positive (in this case 36) then we always choose the positive number to be the principal root. (6). The principal nth root of a real number b, n > 2 an integer, symbolized by n means an = b b a if n is even, a ≥ 0 and b ≥ 0 if n is odd, a, b can be any real number index n radical b radicand Examples Find the principal root: 1. 5 1 2. 811/2 Evaluate 3. (-8)1/3 4. -( 1 1/4 ) 16 Rational Exponents For any nonzero number b, and any integers m and n with n>1, and m and n have no common factors m n b b n m b n m . (Except where b1/n is not a real number) Properties of Powers an = b and Roots a = b1/n for Integer n>0 • The even root of a negative number is not a real number. • Ex: (-9)1/2 is undefined in the real number system Example Evaluate: 4 5 4 5 5 243 3 27 27 5 3 2 3 5 3 27 27 3 81 4 2 3 3 3 27 27 Example Express using rational exponents. 1 4 12 16 4 2 s t 3 4 2s t Express using radicands. 3 4 1 2 5x y 5 x y 3 1 2 4 4 3 5 x y 2 Example Simplify. 7 5 24 r st 3 2 12 r st rs Avoid negative values that would result in an imaginary number! Rational Exponents • bm/n = (b1/n)m = (bm)1/n • b must be positive when n is even. • Then all the rules of exponents apply when the exponents are rational numbers. • Ex: x⅓ • x ½ = • x ⅓+ ½ = • x5/6 • Ex: (y ⅓)2 = y2/3 Radicals • bm/n = (bm)1/n = m and bm/n = (b1/n)m =( Ex: 82/3 = (82)1/3 = = (81/3)2= ( ) m 2 )2 Properties of Radicals • m =( • m ) 2 = • = = • n • n = 3 = a if n is odd = │a │if n is even )2 = 4 =( 2 =6 = = -2 2 (2) = │-2│= 2 Simplifying Radicals • Radicals are considered in simplest form when: • The denominator is free of radicals m • has no common factors between m and n m • has m < n Practice • Simplify • (a1/2b-2)-2 = • = Solving equations 4 5 Solve 616 x 9. 625 x 4 5 isolate the variable 5 4 625 x 5 4 4 5 x 3125 x Calculator: 625^(5÷4) in class assignment • p700 • #4-18 all Sources • http://www.onlinemathtutor.org/help/mathcartoons/mr-atwadders-math-tests/ • images.pcmac.org/SiSFiles/Schools/AL/.../BJHi gh/.../11-1Exponents.ppt