7.1/7.2 Nth Roots and Rational Exponents

How do you change a power to rational form and vice versa?

How do you evaluate radicals and powers with rational exponents?

How do you solve equations involving radicals and powers with rational exponents?

Objectives/Assignment

• Evaluate nth roots of real numbers using both radical notation and rational exponent notation.

• Use nth roots to solve real-life problems such as finding the total mass of a spacecraft that can be sent to Mars.

The Nth root

n a

Index Number

Radicand

Radical

1na

The index number becomes the denominator of the exponent.

n > 1

Radicals

• If n is odd – one real root.

• If n is even and a > 0 Two real roots

a = 0 One real root

a < 0 No real roots

n a

Example: Radical form to Exponential Form

23 x

Change to exponential form.

23x

or

213x

or

1

2 3x

Example: Exponential to Radical Form

23x

Change to radical form.

223 3 or xx

The denominator of the exponent becomes the index number of the radical.

Example: Evaluate Without a Calculator

Evaluate without a calculator.

531. 8 5

33 2

52

32

42. 32 54 2

44 2 2

42 2

Ex. 2 Evaluating Expressions with Rational Exponents

A.

B.

273)9(9 3323

273)9(9 3321

23

Using radical notation

Using rational exponent notation.OR

4

1

2

1

)32(

1

32

132

22552

52

4

1

2

1

)32(

132

2251

52

OR

Example: Solving an equation

Solve the equation:

4 7 9993x 4

4

7 7 9993 7

10000

x

x

44 4 10000

10

x

x

Note: index number is even, therefore, two answers.

Ex. 4 Solving Equations Using nth Roots

A. 2x4 = 162 B. (x – 2)3 = 10

3

81

81

1622

4

4

4

x

x

x

x

15.4

210

102-x

10 2)– (x

3

3

3

x

x

Ex. 1 Finding nth Roots

• Find the indicated real nth root(s) of a.

A. n = 3, a = -125

Solution: Because n = 3 is odd, a = -125 has one real cube root. Because (-5)3 =

-125, you can write:

51253 or 5)125( 31

Ex. 3 Approximating a Root with a Calculator

• Use a graphing calculator to approximate:

34 )5(

SOLUTION: First rewrite as . Then enter the following:

34 )5( 43

5

To solve simple equations involving xn, isolate the power and then take the nth root of each side.

Ex. 5: Using nth Roots in Real Life

• The total mass M (in kilograms) of a spacecraft that can be propelled by a magnetic sail is, in theory, given by:

34

2015.0

fd

mM

where m is the mass (in kilograms) of the magnetic sail, f is

the drag force (in newtons) of the spacecraft, and d is the distance (in astronomical units) to the sun. Find the total mass of a spacecraft that can be sent to Mars using m = 5,000 kg, f = 4.52 N, and d = 1.52 AU.

Solution

The spacecraft can have a total mass of about 47,500 kilograms. (For comparison, the liftoff weight for a space shuttle is usually about 2,040,000 kilograms.

Ex. 6: Solving an Equation Using an nth Root

• NAUTICAL SCIENCE. The Olympias is a reconstruction of a trireme, a type of Greek galley ship used over 2,000 years ago. The power P (in kilowatts) needed to propel the Olympias at a desired speed, s (in knots) can be modeled by this equation:

P = 0.0289s3

A volunteer crew of the Olympias was able to generate a maximum power of about 10.5 kilowatts. What was their greatest speed?

SOLUTION

• The greatest speed attained by the Olympias was approximately 7 knots (about 8 miles per hour).

Rules

• Rational exponents and radicals follow the properties of exponents.

• Also, Product property for radicals

n n na b a b

• Quotient property for radicalsQuotient property for radicalsn

nn

a a

b b

Review of Properties of Exponents from section 6.1

• am * an = am+n

• (am)n = amn

• (ab)m = ambm

• a-m =

•

•

These all work These all work for fraction for fraction

exponents as exponents as well as integer well as integer

exponents.exponents.

ma

1

nmn

m

aa

a

m

mm

b

a

b

a

Ex: Simplify. (no decimal answers)

a. 61/2 * 61/3

= 61/2 + 1/3

= 63/6 + 2/6

= 65/6

b. (271/3 * 61/4)2

= (271/3)2 * (61/4)2

= (3)2 * 62/4

= 9 * 61/2

c. (43 * 23)-1/3

= (43)-1/3 * (23)-1/3

= 4-1 * 2-1

= ¼ * ½

= 1/8

** All of these examples were in rational exponent form to begin with, so the answers should be in the same form!

Try These!

3

1

4

144

23

1

2

1

4

1

2

1

7

7

)32(

)58(

55

Writing Radicals in Simplest Form

4

3

333

32

16

2322754

Example: Using the Quotient Property

Simplify.

416

81

4

44

2 2

3 3

Adding and Subtracting Radicals

Two radicals are like radicals, if they have the same index number and radicand

Example

3 32 and 4 2 are like radicals.

Addition and subtraction is done with like radicals.

Example: Addition with like radicals

Simplify.

4 4 42x x x

Note: same index number and same radicand.

Add the coefficients.

Example: SubtractionSimplify.

5 34x x xNote: The radicands are not the same. Check to see if we can change one or both to the same radicand.

2 3 3

3 3

4

2

x x x x

x x x x

Note: The radicands are the same. Subtract coefficients.

3x x

Writing variable expressions in simplest form

1053

1395 5

8

5

yx

cba