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MATH 250 – REVIEW TOPIC 4
Simplifying Radicals
Have you had any difficulty with radical simplification? Here is a quick
assessment.
Simplify:
a)
4 x^2 c)
4 + 4x + x^2
b)
4 + x^2 d)
4 + 4 tan^2 θ
Answers.
a) 2 |x| Recall: The definition
b) already in simplest form of
x^2 = |x|. If we know
c)
(2 + x)^2 = |2 + x| x ≥ 0, then
x^2 = x.
d)
4(1 + tan^2 θ) =
4 sec^2 θ If we know x < 0, then
= 2| sec θ|
x^2 = −x.
That’s a reasonable start, but it doesn’t meet CALC II demands. Here are
examples of radical simplification that will be used in CALC II.
Ex. 1.
x^2 − x + 1 =
(x^2 − 4 x + 4) =
(x − 2)^2 =
|x − 2 |
Ex. 2.
4 x^2
(1 − x^2 )^2
(1 − x^2 )^2 + 4x^2
(1 − x^2 )^2
1 + 2x^2 + x^4
(1 − x^2 )^2
(1 + x^2 )^2
(1 − x^2 )^2
1 + x^2
| 1 − x^2 |
Ex. 3.
2 x −
8 x
16 x
8 x
64 x 2
64 x^2
(16x 2
2
64 x^2
16 x^2 + 1
| 8 x|
Ex. 4.
9 − 4 x^2 if x =
sin θ; √
sin 2 θ
9 − 9 sin 2 θ =
9(1 − sin 2 θ) =
9 cos^2 θ
= 3| cos θ|
Ex. 5.
1 + [f ′(x)]^2 if f (x) =
(x 2 − 1) 3 / 2 .
First f ′(x) = 2x(x^2 − 1)^1 /^2 , then √ 1 + [f ′(x)]^2 =
1 + 4x^2 (x^2 − 1)
=
4 x^4 − 4 x^2 + 1
=
(2x^2 − 1)^2
= | 2 x 2 − 1 |
All of the radical simplification covered in our examples will be used exten-
sively in evaluating integrals.
Math 250 T4-Simplifying Radicals – Answers Page 4
Answers to Practice Problems
3(x − 1)^2 =
3 |x − 1 |
Return to Problem
9(sec^2 θ − 1) =
9 tan^2 θ = 3| tan θ|
Return to Problem
cos^2 θ
sin 2 θ = | sin θ|
Note: A trig substitution is
necessary to simplify the radical.
Return to Problem
4 sec^2 θ − 4 =
4(sec^2 θ − 1) = 2| tan θ|
Return to Problem
4 x^2 − 1
4 x
16 x^2 + 16x^4 − 8 x^2 + 1
16 x^2
(4x^2 + 1)^2
16 x^2
4 x^2 + 1
| 4 x|
Return to Problem
4.6 f ′ (y) = y 2 −
4 y^2
4 y
4 y^2
16 y^4 + 16y^8 − 8 y^4 + 1
16 y^2
(4y^4 + 1)^2
16 y^2
4 y
| 4 y|
Return to Problem
Beginning of Topic 250 Review Topics 250 Skills Assessment
