7.1/7.2 Nth Roots and Rational Exponents

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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?

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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.

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The Nth root
Radical
Index Number
n gt 1
The index number becomes the denominator of the
exponent.
Radicand
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Radicals

• If n is odd one real root.
• If n is even and
• a gt 0 Two real roots
• a 0 One real root
• a lt 0 No real roots

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Example Radical form to Exponential Form
Change to exponential form.
or
or
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Example Exponential to Radical Form
Change to radical form.
The denominator of the exponent becomes the index
number of the radical.
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Example Evaluate Without a Calculator
Evaluate without a calculator.
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Ex. 2 Evaluating Expressions with Rational
Exponents

• A.
• B.

Using radical notation
Using rational exponent notation.
OR
OR
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Example Solving an equation
Solve the equation
Note index number is even, therefore, two
answers.
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Ex. 4 Solving Equations Using nth Roots

• A. 2x4 162

• B. (x 2)3 10

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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
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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.
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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.
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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.

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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?

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SOLUTION

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

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Rules

• Rational exponents and radicals follow the
  properties of exponents.
• Also, Product property for radicals

• Quotient property for radicals

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Review of Properties of Exponents from section 6.1

• am an amn
• (am)n amn
• (ab)m ambm
• a-m

These all work for fraction exponents as well as
integer exponents.
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Ex Simplify. (no decimal answers)

• (43 23)-1/3
• (43)-1/3 (23)-1/3
• 4-1 2-1
• ¼ ½
• 1/8

• 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

All of these examples were in rational
exponent form to begin with, so the answers
should be in the same form!
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Try These!
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Writing Radicals in Simplest Form
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Example Using the Quotient Property
Simplify.
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Adding and Subtracting Radicals
Two radicals are like radicals, if they have the
same index number and radicand
Example
Addition and subtraction is done with like
radicals.
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Example Addition with like radicals
Simplify.
Note same index number and same radicand. Add
the coefficients.
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Example Subtraction
Simplify.
Note The radicands are not the same. Check to
see if we can change one or both to the same
radicand.
Note The radicands are the same. Subtract
coefficients.
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Writing variable expressions in simplest form