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Algebra 2-nth-root-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?
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
Radical
Index
Number
n
a пЂЅ a
1
n
n>1
Radicand
The index number becomes the
denominator of the exponent.
Radicals
n
a
• If n is odd – one real root.
• If n is even and
a>0
a=0
a<0
Two real roots
One real root
No real roots
Example: Radical form to
Exponential Form
Change to exponential form.
3
x
2
пЂЅ x
2
3
or
пЂЁ пЂ©
пЂЅ x
1
2
3
or
пЂЅ пЂЁx
2
пЂ©
1
3
Example: Exponential to Radical
Form
Change to radical form.
2
x
3
пЂЅ
3
x
2
or
пЂЁ
3
x
пЂ©
2
The denominator of the
exponent becomes the index
number of the radical.
Example: Evaluate Without a
Calculator
Evaluate without a calculator.
1.
пЂЁ
3
пЂ­8
пЂ©
5
пЂЅ
пЂЁ
3
пЂ­2
пЂЅ пЂЁ пЂ­2 пЂ©
пЂЅ пЂ­32
3
5
пЂ©
5
2.
4
32 пЂЅ
пЂЅ
4
4
2
5
2п‚·2
пЂЅ 24 2
4
Ex. 2 Evaluating Expressions
with Rational Exponents
A. 9
3
2
пЂЅ ( 9 ) пЂЅ 3 пЂЅ 27
3
3
1
3
Using radical notation
Using rational exponent
notation.
пЂЅ ( 9 ) пЂЅ 3 пЂЅ 27
1
1
1
B. 32 пЂ­ 2 5 пЂЅ 1 пЂЅ
пЂЅ 2 пЂЅ
2
2
5
2
4
32 5
( 32 )
OR
OR
9
2
32
2
пЂ­2
5
пЂЅ
3
3
1
1
( 32 )
5
2
пЂЅ
1
2
2
пЂЅ
1
4
Example: Solving an equation
Solve the equation:
x пЂ­ 7 пЂЅ 9993
4
x пЂ­ 7 пЂ« 7 пЂЅ 9993 пЂ« 7
4
x пЂЅ 10000
4
4
x пЂЅ
4
4
10000
x пЂЅ п‚± 10
Note: index number
is even, therefore,
two answers.
Ex. 4 Solving Equations Using nth Roots
A. 2x4 = 162
B. (x – 2)3 = 10
2 x пЂЅ 162
(x – 2)  10
x пЂЅ 81
x -2 пЂЅ
x пЂЅ п‚± 81
xпЂЅ
x пЂЅ п‚±3
x п‚» 4 . 15
4
4
4
3
3
3
10
10 пЂ« 2
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:
3
пЂ­ 125 пЂЅ пЂ­ 5
or
( пЂ­ 125 )
1
3
пЂЅ пЂ­5
Ex. 3 Approximating a Root with a
Calculator
• Use a graphing calculator to approximate:
4
( 5)
3
SOLUTION: First rewrite (
enter the following:
4
5)
3
as 5
3
4
. Then
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
2
0 . 015 m
(in kilograms) of the
M пЂЅ
magnetic sail, f is
fd
4
3
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
aп‚·b пЂЅ
aп‚·
n
n
b
• Quotient property for radicals
n
a
b
пЂЅ
n
a
n
b
Review of Properties of Exponents
from section 6.1
•
•
•
•
•
•
am * an = am+n
(am)n = amn
(ab)m = ambm
a-m =
1
a
a
m
a
n
пѓ¦aпѓ¶
пѓ§ пѓ·
пѓЁbпѓё
пЂЅa
m
пЂЅ
m
mпЂ­n
a
m
b
m
These all work
for fraction
exponents as
well as integer
exponents.
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
=Вј
*ВЅ
= 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!
1
1
52 пѓ—54 пЂЅ
1
1
(8 2 пѓ— 5 3 ) пЂЅ
2
(2 пѓ— 3 )
4
7
1
73
4
пЂЅ
пЂ­
1
4
пЂЅ
Writing Radicals in Simplest
Form
3
54 пЂЅ
3
16 пЂЅ
4
32 пЂЅ
3
27 пѓ— 2 пЂЅ 3 2
3
Example: Using the Quotient
Property
Simplify.
4
16
81
пЂЅ
4
2
4
3
4
пЂЅ
2
3
Adding and Subtracting Radicals
Two radicals are like radicals, if they have the same
index number and radicand
Example
3
2 a n d 4 3 2 a re like ra d ic a ls.
Addition and subtraction is done with like radicals.
Example: Addition with like radicals
Simplify.
пЂ­ x пЂ«2 x пЂЅ
4
4
4
x
Note: same index number and same radicand.
Add the coefficients.
Example: Subtraction
Simplify.
4x пЂ­ x
5
x
3
Note: The radicands are not the same. Check to
see if we can change one or both to the same
radicand.
4x п‚· x пЂ­ x
2
2x
3
x пЂ­x
3
x
3
x
3
Note: The radicands are the same. Subtract
coefficients.
x
x
3
Writing variable expressions in
simplest form
5
3
5
9
5
10
5a b c
8x y
13
пЂЅ
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