 # Warm up 1. Change into Scientific Notation 3,670,900,000 Use 3 significant figures 2. Compute: (6 x 10 2 ) x (4 x 10 -5 ) 3. Compute: (2 x 10 7 ) / (8.

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Warm up 1. Change into Scientific Notation 3,670,900,000 Use 3 significant figures 2. Compute: (6 x 10 2 ) x (4 x 10 -5 ) 3. Compute: (2 x 10 7 ) / (8 x 10 3 )

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 a 2 = b a is the cube root of b if a 3 = b Therefore in general we can say: a is an nth root of b if a n = b

Exponents

Properties of Exponents m and n are positive integers a and b are real numbers PropertyDefinitionExample Product a m a n =a m+n (16 3 )(16 7 )=16 10 Power of a power (a m ) n =a mn (9 3 ) 2 =9 6 Power of a quotient (a/b) m =a m /b m (b≠0) (3/4) 5 =3 5 /4 5 Power of a product (ab) m =a m b m (5x) 3 =5 3 x 3 quotient a m /a n =a m-n (a≠0) 15 6 /15 2 =15 4

Example Evaluate:

Example Simplify:

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 b 1/n is an nth root of b. 144 1/2 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: 36 1/2 = 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 means a n = b if n is even, a ≥ 0 and b ≥ 0 if n is odd, a, b can be any real number index radicand radical

Examples Find the principal root: 1. 2.81 1/2 3.(-8) 1/3 4.-( ) 1/4 Evaluate

Rational Exponents For any nonzero number b, and any integers m and n with n>1, and m and n have no common factors (Except where b 1/n is not a real number)

Properties of Powers a n = b and Roots a = b 1/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:

Example Express using rational exponents. Express using radicands.

Example Simplify. Avoid negative values that would result in an imaginary number!

Rational Exponents b m/n = (b 1/n ) m = (b m ) 1/n b must be positive when n is even. Then all the rules of exponents apply when the exponents are rational numbers. Ex: x ⅓ x ½ = x ⅓+ ½ = x 5/6 Ex: (y ⅓ ) 2 = y 2/3

Radicals b m/n = (b m ) 1/n = and b m/n = (b 1/n ) m = ( ) Ex: 8 2/3 = (8 2 ) 1/3 = = (8 1/3 ) 2 = ( ) 2 m m 2

Properties of Radicals = ( ) = ( ) 2 = 4 = = = 6 = = = = a if n is odd = -2 = │a │if n is even = │-2│= 2 m m n n 2 3

Simplifying Radicals Radicals are considered in simplest form when: The denominator is free of radicals has no common factors between m and n has m < n m m

Practice Simplify (a 1/2 b -2 ) -2 = =

Solving equations Solve Calculator: 625^(5÷4) isolate the variable

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