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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. - 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
• 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