MAT 102 - Make Your Own Study Guide – Unit 3
Name _______________________________________
14.1
Date Turned In ___________
Concept – Explain each definition, rule, or process
in this column
Roots and Radicals
Define perfect square.
Show an example of a perfect square.
Define square root.
Show an example of a square root.
What is the difference between the square of a
number and the square root of a number?
The square of 9 is:
Example – Show an example in this column
The square root of 9 is:
Define radical sign.
Show a radical sign:
Define radical.
Show an example of a radical:
Define randicand.
What is the radicand of this radical: √3𝑥
Define principal square root.
√64 =
MAT 102 - Make Your Own Study Guide – Unit 3
14.1
Concept – Explain each definition, rule, or process
in this column
Roots and Radicals (continued)
Define negative square root.
Example – Show an example in this column
−√64 =
Define rational number.
Are the following numbers rational or irrational?
−4
0
2
1
3
Define irrational number.
0.975
1.333̅
𝜋
√6
Define cube root.
Show an example of a cube root.
What is the difference between the cube of a
number and the cube root of a number?
The cube of 27 is:
The cube root of 27 is:
2
MAT 102 - Make Your Own Study Guide – Unit 3
14.1
Concept – Explain each definition, rule, or process
in this column
Roots and Radicals (continued)
Describe how to find the decimal approximation of
square root and cube root on the calculator.
Example – Show an example in this column
Determine the decimal value (rounded to 3 decimal
places) for the following:
√6
3
√9
3
MAT 102 - Make Your Own Study Guide – Unit 3
14.2
Concept – Explain each definition, rule, or process
in this column
Simplifying Radicals
Define index.
Finish these properties of square roots:
Example – Show an example in this column
What is the index of the following:
Simplify:
√4𝑥 =
√𝑎𝑏 =
𝑎
√ =
𝑏
√
Finish this:
𝑦
=
25
Simplify:
√𝑥 2 =
√4𝑦 2 =
Describe how to simplify square root expressions
which have factors with higher powers than 2.
Simplify:
√𝑦 3 =
√𝑥 8 =
√8𝑥 2 𝑦 5 =
4
3
√64
MAT 102 - Make Your Own Study Guide – Unit 3
14.2
Concept – Explain each definition, rule, or process
in this column
Simplifying Radicals (continued)
Finish this:
Example – Show an example in this column
Simplify:
3
3
√8𝑦 3 =
√𝑥 3 =
Describe how to simplify cube root expressions
which have factors with higher powers than 3.
Simplify:
3
√𝑥 4 =
3
√𝑦 9 =
3
√8𝑥 5 𝑦 7 =
Describe how to simplify a radical expression with a
binomial in the numerator (see example).
5
Simplify:
4 + √24
=
2
MAT 102 - Make Your Own Study Guide – Unit 3
14.2
Concept – Explain each definition, rule, or process
in this column
Simplifying Radicals (continued)
What is wrong with this:
4 + √3
= 2 + √3
2
6
Example – Show an example in this column
MAT 102 - Make Your Own Study Guide – Unit 3
14.3
Concept – Explain each definition, rule, or process
in this column
Addition, Subtraction, and Multiplication with
Radicals
Define like radicals.
Show example of like radicals:
Define unlike radicals.
Show example of unlike radicals:
Describe how to add/subtract like radicals.
Add/subtract:
Example – Show an example in this column
2√5 + 6√5 =
√3 − 8√3 =
If you are given an add/subtract with unlike
radicals, what do you do?
Simplify:
5√3 + √27 =
3√32 − √2 =
7
MAT 102 - Make Your Own Study Guide – Unit 3
14.3
Concept – Explain each definition, rule, or process
in this column
Addition, Subtraction, and Multiplication with
Radicals (continued)
Describe how to multiply radicals.
Example – Show an example in this column
Multiply:
√2(√3) =
√𝑥(√𝑥 − 4) =
(√𝑦 + 7)(√𝑦 − 2) =
8
MAT 102 - Make Your Own Study Guide – Unit 3
14.4
Concept – Explain each definition, rule, or process
in this column
Rationalizing Denominators
Describe how to rationalize fraction with square
root monomial in denominator.
Example – Show an example in this column
Simplify:
2
√𝑥
Describe how to rationalize fraction with cube root
monomial in denominator.
=
Simplify:
2
3
√𝑥
=
2
3
√𝑥 5
9
=
MAT 102 - Make Your Own Study Guide – Unit 3
14.4
Concept – Explain each definition, rule, or process
in this column
Rationalizing Denominators (continued)
Define conjugates.
Example – Show an example in this column
What is the conjugate of each of these?
3 − √𝑥
−4 + √𝑥
Describe how to rationalize fraction with binominal
(containing square root radical) in denominator.
Simplify:
7
5 − √𝑥
10
=
MAT 102 - Make Your Own Study Guide – Unit 3
14.5
Concept – Explain each definition, rule, or process
in this column
Equations with Radicals
Finish out this Property of Real Numbers:
Example – Show an example in this column
Use this property to solve this equation (by
squaring both sides of the equation):
If 𝑎 = 𝑏, then 𝑎2 =
√2𝑥 − 3 = 5
Describe how to solve equation with radical
expression.
Solve:
𝑎 = 3 + √3𝑎 − 9
Define extraneous solution.
Why you must check your solutions when you have an equation with a radical?
11
MAT 102 - Make Your Own Study Guide – Unit 3
14.5
Concept – Explain each definition, rule, or process
in this column
Equations with Radicals (continued)
Example of problem with extraneous solution:
Example – Show an example in this column
Solve the equation. Check all of your solutions. Are
any of them extraneous?
√18 − 𝑥 + 6 = 𝑥
Describe how to solve an equation with a cube root
radical in it?
Solve:
3
√3𝑥 + 4 + 2 = 4
12
MAT 102 - Make Your Own Study Guide – Unit 3
14.5
Concept – Explain each definition, rule, or process
in this column
Equations with Radicals (continued)
Describe how to solve an equation with 2 radicals in
it (and no other terms).
Describe how to solve an equation with 3 terms,
two of which are radicals.
13
Example – Show an example in this column
Solve:
√−2𝑦 + 2 = √2𝑦 + 6
Solve:
√𝑥 + 1 + √𝑥 − 19 = 10
MAT 102 - Make Your Own Study Guide – Unit 3
14.6
Concept – Explain each definition, rule, or process
in this column
Rational Exponents
What does 𝑥
1⁄
𝑛
Example – Show an example in this column
Simplify:
mean?
27
What does 𝑥
𝑚⁄
𝑛
1⁄
3
Simplify:
mean?
64
Finish this summary of exponent rules:
2⁄
3
𝑎1 =
𝑥1 =
Exponent of 0:
𝑎0 =
𝑥0 =
Product Rule:
𝑎𝑚 ∙ 𝑎𝑛 =
𝑎𝑚
𝑎𝑛
Negative Exponents:
Power Rule:
𝑥3 ∙ 𝑥7 =
𝑥 11
=
𝑥6
=
1
𝑎−𝑛
𝑎−𝑛 =
=
𝑥 −3 =
(𝑎𝑚 )𝑛 =
Power Rule for Products:
Power Rule for Fractions:
(𝑎𝑏)𝑛
=
Simplify:
Exponent of 1:
Quotient Rule:
=
1
=
𝑥 −5
=
𝑎 𝑛
(𝑥 3 )4 =
(𝑏 ) =
(2𝑥)4 =
𝑥 3
( ) =
5
14
MAT 102 - Make Your Own Study Guide – Unit 3
14.6
Concept – Explain each definition, rule, or process
Example – Show an example in this column
in this column
Rational Exponents (continued)
Simplify these expressions using exponent rules. Leave your answer in exponential form.
𝑥
1⁄
3
𝑥
1⁄
6
2
𝑥 ⁄3
∙𝑥
=
=
1⁄
5
𝑥−
1
2⁄
3
𝑥−
2⁄
5
=
=
1⁄
2⁄
4
5
(𝑥 )
=
3⁄
1⁄
4
3
(2𝑥 )
=
𝑥 1⁄2
( ) =
8
15
MAT 102 - Make Your Own Study Guide – Unit 3
14.6
Concept – Explain each definition, rule, or process
Example – Show an example in this column
in this column
Rational Exponents (continued)
Simplify by changing first to exponential form. Then simplify and express your answer in radical form.
5
√ 3√𝑥 =
16
MAT 102 - Make Your Own Study Guide – Unit 3
14.7
Concept – Explain each definition, rule, or process
in this column
Functions with Radicals
Define radical function.
Show an example of a radical function.
Define relation.
Show an example of a relation.
Define domain.
What is the domain of this relation:
{(2,1), (4,5), (6,1)}
Define function.
Are the following relations functions? Explain.
Example – Show an example in this column
{(2,1), (4,5), (6,1)}
{(−1,2), (2,5), (−1,4)}
Define vertical line test.
Is this relation a function? Explain.
17
MAT 102 - Make Your Own Study Guide – Unit 3
14.7
Concept – Explain each definition, rule, or process
in this column
Functions with Radicals (continued)
Describe how to find the domain for a radical
function with square root.
Example – Show an example in this column
What is the domain of the following radical
function?
𝑓(𝑥) = √3𝑥 − 1
Describe how to find the domain for a radical
function with cube root.
What is the domain of the following radical
function?
3
𝑔(𝑥) = √𝑥 + 1
Describe how to graph radical function with a
square root.
Graph the following radical function:
𝑓(𝑥) = √𝑥 − 1
18
MAT 102 - Make Your Own Study Guide – Unit 3
14.7
Concept – Explain each definition, rule, or process
in this column
Functions with Radicals (continued)
Describe how to graph radical function with a cube
root.
19
Example – Show an example in this column
Graph the following radical function:
3
𝑔(𝑥) = √𝑥 + 1