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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.

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7 1 7 2 nth roots and rational exponents

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
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
Example: Radical form to Exponential Form

Change to exponential form.



example exponential to radical form
Example: Exponential to Radical Form

Change to radical form.

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

example evaluate without a calculator
Example: Evaluate Without a Calculator

Evaluate without a calculator.

ex 2 evaluating expressions with rational exponents
Ex. 2 Evaluating Expressions with Rational Exponents



Using radical notation

Using rational exponent notation.



example solving an equation
Example: Solving an equation

Solve the equation:

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

ex 1 finding nth roots
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:


ex 3 approximating a root with a calculator
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
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.


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
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?

  • The greatest speed attained by the Olympias was approximately 7 knots (about 8 miles per hour).
  • Rational exponents and radicals follow the properties of exponents.
  • Also, Product property for radicals
  • Quotient property for radicals
review of properties of exponents from section 6 1
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.

ex simplify no decimal answers
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

Ex: Simplify. (no decimal answers)

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

adding and subtracting radicals
Adding and Subtracting Radicals

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


Addition and subtraction is done with like radicals.

example addition with like radicals
Example: Addition with like radicals


Note: same index number and same radicand.

Add the coefficients.

example subtraction
Example: Subtraction


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.