3-4: Rational 3-4: Rational Exponents and Exponents and

Radical EquationsRadical Equations

3-4: Rational 3-4: Rational Exponents and Exponents and

Radical EquationsRadical EquationsEnglish CasbarroEnglish Casbarro

Unit 3Unit 3

Roots of nth roots

Case Roots Example

Odd index 1 real root

Even index (positive radicand)

2 real roots

Even index (negative radicand)

No real roots

radicand of 0 1 root of 0

2=8;2=8 33

2±=164

i2±=164

0=0

Find all real roots

1. 2. 3.

4. 5. 6.

4 81 3 125 6 729

4 256 6 1 3 64

Simplifying nth roots

Recall: and The same applies here:

Ex.

nnn baab =)(n

nn

ba

ba

=)(

baab •=

3333 22=2•8=16

33 93 23 7 ==• xxxx

Definition of a rational exponent

A rational exponent is defined as:

Where m and n are integers and n ≠ 0.

Ex.

nm

n m aa =

2=16=16 441

( ) 4=2=8=8 22332

Write each expression in radical form and simplify.

Write each expression using rational exponents.

Solve.

Turn in the following problems1. The formula , known as Kleiber’s

Law, relates the metabolism rate P of an

organism in Calories per day and the body

mass m of the organism in kilograms. The

table shows the typical body mass of several

members of the cat family.

Animal Mass(kg)

House cat

4.5

Cheetah 55.0

Lion 170.0

a. What is the metabolism rate of a cheetah to the nearest Calorie per day?b. Approximately how many more Calories of food does a lion need to consume each day than a house cat does?

2. For a pendulum with length L, in meters, the expression models the time in seconds for the pendulum to complete one full swing. In this expression, g is acceleration due to gravity, 9.8 m/s2. a. Simplify the expression by rationalizing the denominator. b. To the nearest second, how long does it take for a pendulum with a length of 3.5 m to complete one full swing?3. The surface area S, of a sphere with volume, V, is . What effect does increasing the volume of a sphere by a factor of 8 have on its surface area?

A. The surface area doubles.B. The surface area triples.C. The surface area increases by a factor of 4.D. The surface area increases by a factor of 8.

Like radicals are radicals that have the same radicands and indexes.

You must simplify before adding and subtracting like radicals.

Remember that imaginary numbers also have conjugates– they allow you torationalize denominators and expressions.

Radical Equations

The entire point of solving the equation is to isolate the radical.

Turn in the following problemsMatch each function to its graph.

1.2.3. 4.

5. The formula, , approximates thevelocity in miles per hour necessary to escape the

gravity of a planet with acceleration due to gravity g in ft/s2 and radius R in miles.On Earth, which has a radius of 3960 mi, the acceleration due to gravity is 32 f/s2

On the moon, which has a radius of 1080 mi, the acceleration due to gravity is

about that on Earth. How much faster would a vehicle need to be travelling toescape Earth’s gravity than the moon’s gravity?

6. For a pendulum with length L in meters, the function describes the time in seconds.

a. Find the length of the pendulum that completes one complete swing in 2.2 seconds. b. A clockmaker needs a pendulum to complete 120 complete swings in one minute. To the nearest hundredth of a meter, how long should the

pendulum be?