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)

Lesson1.7 Rational ExponentsObjective:

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

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 means an = b

if n is even, a ≥ 0 and b ≥ 0

if n is odd, a, b can be any real number

n bindex

radicand

radical

n b a

Examples

5 1

Find the principal root:

1.

2. 811/2

3. (-8)1/3

4. -( ) 1/41

16

Evaluate

Properties of Powers an = b and Roots a = b1/n for Integer n>0• Any power of a real number is a real number.

• Ex: 43 = 64 (-4)3 = -64

• The odd root of a real number is a real .• Ex: 641/3 = 4 (-64)1/3 = -4

• A positive power or root of zero is zero.• Ex: 0n = 0 0 1/n = 0

Properties of Powers an = b and Roots a = b1/n for Integer n>0• A positive number raised to an even power

equals the negative of that number raised to the same even power.• Ex: 32 = 9 (-3)2 = 9

• The principal root of a positive number is a positive number.• Ex: (25)1/2 = 5

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

Warm up

• 1.

• 2. (x1/3y-2)3

• 3.

4 1

169

Practice

• (-49)1/2 =

• (-216)1/3 =

• -( 1/81)1/4 =

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

Practice

• 274/3 =

• (a1/2b-2)-2 =

Radicals• is just another way of writing b1/2.

• The is denoting the principal (positive) root

• is another way of writing b1/n , the principal nth root of b.

• so:

• = b1/n = a where an = 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 an = b

Radicals• bm/n = (bm)1/n =

and

bm/n = (b1/n)m = ( )

Ex: 82/3 = (82)1/3 =

= (81/3)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 =

4

1 1 (2x)3/2 = 3

3

1 = y– 4/7

y4/7

Properties of Radicals• = ( ) = ( )2 = 4

• = = = 6

• = = =

• = a if n is odd = -2

• = │a │if n is even = │-2│= 2

mm

n

n

2

3

2

Simplify• =

• =

• =

• =

3

6

Warm up

2

3

36• 1.

• 2.

• 3.

3

4

27

4

3

81

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• -9xy3

• -6

+

• 4

-

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.

• ● =

n a m b

5 xy 5 xy2 5 xy23

Operations with Radicals

• Simplify:

332 xyxy 2 2 2

Sources• http://www.onlinemathtutor.org/help/math-

cartoons/mr-atwadders-math-tests/