Chapter 5: Exponential and Logarithmic Chapter 5: Exponential and Logarithmic FunctionsFunctions5.1: Radicals and Rational Exponents5.1: Radicals and Rational ExponentsEssential Question: Explain the meaning of

using radical expressions

mna

5.1: Radicals and Rational 5.1: Radicals and Rational ExponentsExponentsBACKGROUND INFO (NO NEED TO COPY)Recall that when x2 = c (some constant),

there were two solutions, and , when the constant was positive. You had no solutions when the constant was negative.◦ x2 = 9 → x = 3 or x = -3

When x3 = c, there was only one solution, , and the answer was positive or negative depending on if x was positive or negative to start.◦ x3 = 64 → x = 4◦ x3 = -64 → x = -4

All other higher roots act in a similar fashion

x x

3 x

5.1: Radicals and Rational 5.1: Radicals and Rational ExponentsExponentsSolutions to xn = c

◦When n (exponent) is even If c > 0, one positive and one negative

solution If c = 0, one solution (x = 0) If c < 0, no solution

◦When n is odd One solution

Recall help:◦Even # power = even # of roots◦Odd # power = odd # of roots

5.1: Radicals and Rational 5.1: Radicals and Rational ExponentsExponentsnth roots

◦The nth root of c is denoted by either of the symbols:

Rules about nth roots◦If the outside root is the same,

numbers underneath can be multiplied and divided

◦If the number underneath the root is the same, they are like terms, and can be added or subtracted

1

or nn c c

5.1: Radicals and Rational 5.1: Radicals and Rational ExponentsExponentsExample 1: Operations on nth roots

◦ ◦ ◦ ◦

Example 2: Evaluating nth roots◦ Use a calculator to approximate each

expression the nearest ten thousandth.

◦ , entered as “40^(1/5)” ≈ 2.0913◦ , entered as “225^(1/11)” ≈ 1.6362

1540111225

8 12

12 756 43 8x y

(5 )(5 )c c

96 16 46 6

4 3 25 3 35 32 3 3 23 3 33 33 3 28 x x y x y yy

225 55 5 2c c cc

5.1: Radicals and Rational 5.1: Radicals and Rational ExponentsExponentsRational Exponents

◦Rational exponents of the form are called nth roots. Rational exponents can also be written in the form .

◦The numerator of the exponent represents the power a base is taken to.

◦The denominator of the exponent represents the root.

◦The order of application does not matter◦

1nc

mnc

3 1 12 2 234 (4 ) 64 64 8 3 12 2 3 3 34 (4 ) ( 4) 2 8

5.1: Radicals and Rational 5.1: Radicals and Rational ExponentsExponentsAssignmentPage 334

◦Problems 1 – 37, odd problems Ignore the instructions about not using a

calculator in problems 1 – 15 Make sure to simplify your square roots Show all non-calculator work Due tomorrow

Chapter 5: Exponential and Logarithmic Chapter 5: Exponential and Logarithmic FunctionsFunctions5.1: Radicals and Rational Exponents 5.1: Radicals and Rational Exponents (Day 2)(Day 2)Essential Question: Explain the meaning of

using radical expressions

mna

5.1: Radicals and Rational 5.1: Radicals and Rational ExponentsExponentsLaws of Exponents (A recap)

◦crcs = cr+s

Multiplying same base == add exponents◦ = cr-s

Dividing same base == subtract exponents◦(cr)s = crs

Power to power == multiply exponents◦(cd)r = crdr and

Power on outside == multiply exponents to all bases

◦c-r = Negative exponents move to other side of

the fraction and become positive

r

s

c

c

r r

r

c c

d d

1rc

5.1: Radicals and Rational 5.1: Radicals and Rational ExponentsExponentsSimplifying Expressions with

Rational Exponents (Ex 3)◦ Write the expression using only positive

exponents

◦ Distribute the 2/3 on the outside

◦ Simplify the coefficient part & move

negative exponents

23 3

348r s

22 2 23 3 13 3 33 2 23 3 34 4 28 8 8r s r s r s

1

12

23 2 2

2

48

r

sr s

5.1: Radicals and Rational 5.1: Radicals and Rational ExponentsExponentsSimplifying Expressions with Rational

Exponents◦ Example 4:

◦ Distribute… and since bases are shared, add the exponents

◦

◦ Example 5: ◦ Take the -2 power first, then add exponents

to like bases (as in Ex 4 above)◦

1 3 3

2 4 2x x x

1 3 1 3

2 4 2

5242x x x x

5 72 4

24x y xy

5 7 5 72 4 2

12 22

124 4 2x y xy x y yx y x

5.1: Radicals and Rational 5.1: Radicals and Rational ExponentsExponentsSimplifying Expressions with Rational

Exponents◦Ex 6: Write the expression without using

radicals, using only positive exponents◦ ◦Get rid of the radicals. Root =

denominator◦ ◦Multiply powers of powers.◦ ◦Add exponents of common bases◦

1210 5k kc c

11 1 1 22 10 210 5 5( ) ( )k k k kc c c c

11 1 210 2 2 45( ) ( )

k kk kc c c c

22 4 44 4k k k k k

c c cc c

5.1: Radicals and Rational 5.1: Radicals and Rational ExponentsExponentsRationalizing the Numerator/Denominator

◦ Rationalizing means removing all roots from the specified side of a fraction

◦ Simple roots can be removed by multiplying top/bottom by the root.

◦

◦ Complex roots can be removed by multiplying with the conjugate

◦

◦ Rationalizing a numerator works the same as above

7 5

5 5

7 5

5

2 3 6 6 2 6

9 63 6 3

2

36

6 6

5.1: Radicals and Rational 5.1: Radicals and Rational ExponentsExponentsAssignmentPage 334

◦Problems 39-77, odd problems Make sure to simplify your square roots Show all non-calculator work Due whenever we get back