7.1/7.2 Nth Roots and Rational Exponents

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# 7.1/7.2 Nth Roots and Rational Exponents - PowerPoint PPT Presentation

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.

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### 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

Index Number

n > 1

The index number becomes the denominator of the exponent.

• 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

Example: Radical form to Exponential Form

Change to exponential form.

or

or

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

Example: Evaluate Without a Calculator

Evaluate without a calculator.

Ex. 2 Evaluating Expressions with Rational Exponents

A.

B.

Using rational exponent notation.

OR

OR

Example: Solving an equation

Solve the equation:

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

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:

or

Ex. 3 Approximating a Root with a Calculator
• Use a graphing calculator to approximate:

SOLUTION: First rewrite as . Then enter the following:

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:

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
• Also, Product property for radicals
Review of Properties of Exponents from section 6.1
• am * an = am+n
• (am)n = amn
• (ab)m = ambm
• a-m =

These all work for fraction exponents as well as integer exponents.

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

(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!

Example

Simplify.

Note: same index number and same radicand.