Download Rational Exponents / Radical Expressions

§7.1 / 7.2 Assignment
Notes
1. Exercises from 7.1, 7.2
2. Nov 13 - Quiz #7: 6.6, 7.1, 7.2, Cumulative section:
6.1 - 6.3
3. Nov 20 - Exam #3: 6.1 - 6.3, 6.6, 7.1 - 7.4, 7.6; and
Cumulative portion
Study guide: Chapter 6 test # 1 - 12, 16 - 21; Chapter 7
test 1 - 13, 17 - 21;
Cumulative Test: Chapters 5-7 #1 - 18, 23 - 31, 36 - 39
(Math 1010)
M 1010 §7.1/7.2
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§7.1 - Radicals and Rational Exponents
Notes
√
n-th roots of numbers ( n ) are equal value factors of a number, n-many
times.
Example Because 4 · 4 · 4 = 64, the number 4 is the 3rd root, or cube
root, of 64.
The principal root of a number has the same sign as that number.
√
Example The principal square root of 49 is 7. We say 49 = 7. This
avoids ambiguity because (−7) is also a square root of 49, but it is not the
principal square root because it does not have the same sign as 49.
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§7.1 - Radicals and Rational Exponents
Notes
When does a real number have an nth root, and how many can it have?
I
If real number a is positive and n is even, then a has two real nth
√
√
roots, n a and − n a.
I
If a is any real number and n is odd, then a has only one real root,
√
n
a.
I
If real number a is negative and n is even, then a has no real nth root.
Perfect squares and cubes have rational square or cube roots, respectively.
This pertains also to perfect nth roots.
(Math 1010)
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§7.1 - Radicals and Rational Exponents
Notes
It is useful to review page 300. Inverse properties of nth powers and nth
roots:
√
1. If a has a principal root, then ( n a)n = a.
√
2. If n is odd, then n an = a.
√
3. If n is even, then n an = |a|.
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§7.1 - Radicals and Rational Exponents
a1/n =
√
n
Notes
a
Again, study page 300 and compare to page 453:
1. ar · as = ar +s
ar
2. s = ar −s
a
3. (ab)r = ar · b r
4. (ar )s = ars
a r
ar
5.
= r
b
b
6. a0 = 1
7. a−1 = a1r
a −1 b r
8.
=
b
a
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§7.1 - Radicals and Rational Exponents
Notes
√
3
# 1: 91 Rewrite using rational exponents: x x 6 .
√
4
t
# 2: 97 Rewrite using rational exponents: √
t5
# 3: 113 Simplify (c 3/2 )1/3
!3
x 1/4
# 4: 119 Simplify
x 1/6
q
4 √
# 5: 123 Simplify
x3
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§7.2 - Simplifying Radical Expressions
Notes
Simplyfing a radical.
√
√
√ √
12 = 4 · 3 = 4 · 3. Why do these last equalities work?
There are product and quotient rules for radicals that mimic the product
and quotient rules for exponents:
√ √
√
1. n u · v = n u · n v
r
√
n
u
u
2. n
= √
n
v
v
We will use this rule to remove perfect squares from radicals. First we
remove constants, if able. Then we remove variable factors, if able.
M 1010 §7.1/7.2
(Math 1010)
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§7.2 - Simplifying Radical Expressions
Notes
Removing constants.
# 1: 1 Simplify the radical.
# 2: 5 Simplify the radical.
√
18.
√
96.
√
# 3: 10 Simplify the radical. 1176.
√
# 4: 31 Simplify the radical. 3 112.
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§7.2 - Simplifying Radical Expressions
Notes
Removing variable factors.
√
# 1: 19 Simplify 9x 5 .
√
3
# 2: 33 Simplify 40x 5
r
2
4 3u
# 3: 52 Simplify
16v 8
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§7.2 - Simplifying Radical Expressions
Notes
Rationalizing Denominators
When simplifying, we want radicals to contain no fractions, and
denominators to contain no radicals.
r
√
√ √
√
3
3
3
5
15
Example:
=√ =√ ·√ =
5
5
5
5
5
We multiply the fraction with 1/1, written as equal factors to the
denominator over equal factors, and enough to cancel out the radical.
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§7.2 - Simplifying Radical Expressions
Notes
Rationalize the denominator:
r
1
#1 : 55
3
s
1
#2 : 63
y
#3 : 69 √
6
3b 3
s
2x
#4 : 71 3
3y
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Notes