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

Lesson1.7 Rational Exponents Objective:

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

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 /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: /2 3.(-8) 1/3 4.-( ) 1/4 Evaluate

Properties of Powers a n = b and Roots a = b 1/n for Integer n>0 Any power of a real number is a real number. Ex: 4 3 = 64 (-4) 3 = -64 The odd root of a real number is a real. Ex: 64 1/3 = 4 (-64) 1/3 = -4 A positive power or root of zero is zero. Ex: 0 n = 0 0 1/n = 0

A positive number raised to an even power equals the negative of that number raised to the same even power. Ex: 3 2 = 9 (-3) 2 = 9 The principal root of a positive number is a positive number. Ex: (25) 1/2 = 5

The even root of a negative number is not a real number. Ex: (-9) 1/2 is undefined in the real number system

Warm up (x 1/3 y -2 ) 3 3.

Practice (-49) 1/2 = (-216) 1/3 = -( 1/81) 1/4 =

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

Practice 27 4/3 = (a 1/2 b -2 ) -2 =

Radicals is just another way of writing b 1/2. The is denoting the principal (positive) root is another way of writing b 1/n, the principal nth root of b. so: = b 1/n = a where a n = b if n is even and b< 0, is not a real number; if n is even and b≥ 0, is the nonnegative number a satisfying a n = b

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

Practice Change from radical form to rational exponent form or visa versa. (2x )-3/2 x>0 = (-3a) 3/7 = 1 = (2x) 3/2 = = y – 4/7 y 4/7

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 2

Simplify = 3 6

Warm up

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

Rationalizing the Denominator To rationalize a denominator multiply both the numerator and denominator by the same radical. Ex: ● = = 2

Rationalizing the Denominator If the denominator is a binomial with a radical, it is rationalized by multiplying it by its conjugate. ( The conjugate is the same expression as the denominator but with the opposite sign in the middle, separating the terms. (√m +√n)(√m -√n)= m-n

Rationalizing the Denominator -9xy

Operations with Radicals Adding or subtracting radicals requires that the radicals have the same number under the radical sign (radicand) and the same index. + =

Operations with Radicals Multiplying Radicals and can only be multiplied if m=n. ● = 223

Operations with Radicals Simplify: 2 22

Sources cartoons/mr-atwadders-math-tests/