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Unit 9
Radical Expressions and Equations
Introduction
The real numbers are made up of many types of numbers. The simplest set of
numbers is called the Natural Numbers. These numbers be can written as the list given
below.
Natural Numbers = { 1, 2, 3, 4, … }
There is also a set of numbers called the Integers. These numbers can also be shown
as a list.
Integers = { 0, 1, -1, 2, -2, 3, -3, … }
Another set of numbers is called the Rational Numbers because they can be represented
as fractions or ratios. This set of numbers can also be described as all the terminating or
repeating decimal numbers. Some examples of values from this set are given below.
5
= 1.25
4
This is a terminating decimal value. ( Terminating means it stops. )
1
= 1.33333… = 1.3
3
2
= .18181818… = .18
11
This is a repeating decimal value.
This is also a repeating decimal value.
1
= .142857142857… = .142857
7
Again we have a repeating decimal value.
The set of numbers we are going to study in this unit is called the Irrational Numbers.
This set of numbers is the opposite of the Rational Numbers. The Irrational numbers
are decimal values that do not terminate and do not repeat. Below are some examples
of these numbers.
 = 3.1415926535897932384626 … The value of pi is irrational, it will never
stop and never form a pattern in order
to repeat.
Any square root that does not terminate will be an irrational number.
2 = 1.41421356 … This value will never terminate and never form a repeating
pattern. So the 2 , like most square roots, is irrational.
.
Unit 9
Vocabulary and Concepts
Index
The index of a radical tells us the number of times a factor must be used
to generate the radicand.
Radical
A radical is the symbol
which tells us to reverse a square. The
understood index here is 2.
Radicand
The radicand is the value or expression inside the radical.
Key Concepts for Exponents and Polynomials
Simplifying Radicals
Multiplying Radicals
To simplify radicals means to change any perfect squares into
their roots. If some factors of the radicand are perfect
squares and some are not, then we split the radical into
two roots - one with the perfect squares, and one with the
factors that are not perfect squares.
To multiply radicals or square roots we multiply the radicands
and simplify the radical.
Adding/Subtracting Radicals
Dividing Radicals
To add/subtract radicals we add or subtract the
coefficients of the radicals. This can only be done
when the radicands and indices are the same.
To divide radicals or square roots we divide out the common
factors of the radicands and simplify the radicals and rationalize
the denominator where necessary.
Rationalizing the Denominator
To rationalize a denominator we multiply the
fraction by the denominator over itself.
Unit 9 Section 1
Objective

The student will multiply radicals.
In order to work with radicals we should first make sure we can use the vocabulary
associated with these numbers. The expression below shows us a radical value and
the terms we use for it.
Index
3
x
Radicand
Radical
The radical is the symbol in the example. This symbol means we are reversing a
square, cube, or 4th power.
The radicand is the number or expression inside the radical.
The index tells us how many times a factor must be used to multiply to the number
or expression in the radical to produce the radicand.
Below are some examples of radical expressions and their meanings.
25
Find the number that times itself equals twenty-five. This evaluates to 5
because 52 = (5)(5) = 25.
3
64
Find the number that used as a factor three times equals 64. This evaluates
to 4 because 43 = (4)(4)(4) = 64.
4
81
Find the number that used as a factor four times equals 81.
to 3 because 34 = (3)(3)(3)(3) = 81.
This evaluates
In these examples we must emphasize that a radical without an index is understood
to be a square root. This means that the understood index is ‘2’. In Algebra I we will
be primarily concerned with the square roots.
When we work with radicals or square roots we must understand the difference between
the exact value of a number and an approximation of a number. The square root of ‘2’, when
written with a radical is exact; when written as a decimal, it is rounded off and so is an
approximation.
Exact
Approximation
2
1.41421
It will not matter how many decimal places we use with the value on the right it will always
be an approximation.
To find the exact answers we will need to be able to perform operations with radicals or
square roots. We will also at times have variables and expressions inside the radical.
When there are expressions inside a radical we cannot find an approximation and we
must be able to do operations with the radicals themselves. The first operation we
need to explore is multiplication of square roots. We will examine some problems to
develop a method for multiplication.
Example A
4 .
9
This problem can be done with two different methods.
Method 1
4 .
Example B
Method 2
4 .
9
9
2 . 3
36
6
6
25 .
16
This problem can be done with two different methods.
Method 1
25 .
Example C
16
Method 2
25 .
16
5 . 4
400
20
20
4 .
25
This problem can be done with two different methods.
Method 1
4 .
25
Method 2
4 .
2 . 5
100
10
10
25
From examples A through C we can see that we can multiply the radicands, the numbers
in the radicals, first and then take the square roots afterwards if we want. There is one
restriction that we need to know. We can only multiply radicals that have the same
index. We will normally only be using square roots.
Below are some examples of multiplying square roots and simplifying when possible.
Example A
a .
Example B
2x .
a3
3a .
18 x
a4
36x 2
a2
6x
Since (a2)(a2) = a4
the square root of
a4 must be a2.
Example C
Example D
3xy .
5b
5y 5
225 x 2 y 6
15ab
Since there are no
perfect squares in the
radicand we cannot
simplify this answer.
Since (6x)(6x) = 36x2
the square root of
36x2 must be 6x.
15 x .
15xy3
Since (15xy3)(15xy3)
is equal to 225x2y6
the square root of
225x2y6 must be 15xy3.
VIDEO LINK: Khan Academy Multiply and Simplify a Radical Expression 1
Exercises Unit 9 Section 1
Find the square roots. If there is no real answer then write “no real number”.
1.
2.
81
4
3.
36
4. – 144
Estimate the square roots to the nearest tenth. Check your answers by multiplying
your estimate times itself. Show your work.
5.
6.
14
7.
125
8.
91
190
Evaluate the following. Show your work.
9. Evaluate
3
10. Evaluate
-
11. Given
x2 –4
1
2
with x = 14
6x  2 + 3
f(x) =
4x  3
with x = 11
Find f(7) and f(21), show your work.
Multiply the following and simplify where possible.
12.
12 .
15.
5 .
7
16.
( x  1) .
18.
3 .
48
19.
2x 3 .
21.
8 .
18
22.
2 .
5 .
24.
8xy 4 .
2xy 2
25.
7 .
27.
ab 5 c 4 .
a 3b 3 c 2
28.
30.
ab .
31.
13.
3
xy
2 .
14.
x .
x
17.
7 .
28
20.
45 .
5
10
23.
12.5 .
14 .
2
26.
11 .
3
3a .
2 .
6a
29.
13 .
13
2w .
98w
32.
4 .
100
50
( x  1)
32 x
2
Unit 9 Section 2
Objective

The student will simplify square roots.
Simplifying square roots requires that we recognize perfect squares. The list below
contains all the perfect squares from 1 to 225. These are the squares of the natural
numbers from 1 to 15.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Variable expressions can also be perfect squares. The list below contains examples
of these expressions.
x2 = (x)(x), x4 = (x2)(x2), x6 = (x3)(x3), x8 = (x4)(x4), x10 = (x5)(x5)
Any variable with an even exponent is a perfect square.
In the last section we were asked to multiply and simplify square roots. The radicals
that we were asked to simplify usually contained only perfect squares. These would
be problems similar to the ones shown below.
16
49 y 2
25a 2 b 4
( x  4) 2
4
7y
5ab 2
( x  4)
There were also a few problems that we could not simplify such as those given below.
17
x
2z
5xyz
These problems have no perfect squares. There are problems that have both types
of factors in the radicand. When we have factors that are perfect squares, then
simplifying the radical requires that we simplify the radical as much as possible. This
will mean splitting the radical into factors of perfect and non-perfect squares. The
examples below show us this process.
Example A
Simplify
75
The value 75 is not a perfect square. However, 75 can be factored into 25 . 3,
and 25 is a perfect square. This means we can write the 75 in factored form.
75
25  3
25
5.
.
We factored the radicand.
3
3
Since 25 . 3 = 75 we can write this as a product of two radicals.
The square root of 25 is 5 so we simplify the first square root.
5 3 We don’t have to write the dot for multiplying so this is the best form.
Example B
Simplify
44
The value 44 is not a perfect square. However, 44 can be factored into 4 . 11,
and 4 is a perfect square. This means we can write the 44 in factored form.
44
We factored the radicand.
4  11
4 .
11
Since 4 . 11 = 44 we can write this as a product of two radicals.
2.
11
The square root of 4 is 2 so we simplify the first square root.
2 11
We don’t have to write the dot for multiplying so this is the best form.
In these first two examples we put the perfect square in the first radical each time. This
makes our simplifying a little easier. When we split the radical into the product of two
radicals we will always put the perfect squares into the first radical. We should always
find the largest perfect square that is a factor as well.
Example C
Simplify
128
The value 128 is not a perfect square. However, 128 can be factored into 64 . 2,
and 64 is the largest perfect square. So we can write the 128 in factored form.
128
64  2
We factored the radicand.
64 .
2
8.
2
8
The square root of 64 is 8 so we simplify the first square root.
We don’t have to write the dot for multiplying so this is the best form.
2
Example D
Since 64 . 2 = 128 we can write this as a product of two radicals.
Simplify
8x 7
The value 8x7 is not a perfect square. But, 8x7 can be factored into 4x6 . 2x,
and 4x6 is a perfect square. This means we can write the 8x7 in factored form.
8x 7
4x 6  2x
4x 6
2x3 .
.
2x
2x
2x3 2 x
We factored the radicand.
Since 4x6 . 2x = 8x7 we can write this as a product of two radicals.
The square root of 4x6 is 2x3 so we simplify the first square root.
We don’t have to write the dot for multiplying so this is the best form.
Example E
Simplify
98 xy 2 z 5
The value 98xy2z5 is not a perfect square. However, 98xy2z5 can be factored
into 49y2z4 . 2xz, and 49y2z4 is a perfect square. This means we can write the
98xy2z5 in factored form.
98 xy 2 z 5
49 y 2 z 4  2 xz
49 y 2 z 4 .
2 xz Since 49y2z4 . 2xz = 98xy2z5 this can be the product of two radicals.
7yz2 .
7yz2
We factored the radicand.
2 xz The square root of 25 is 5 so we simplify the first square root.
2 xz We don’t have to write the dot for multiplying so this is the best form.
VIDEO LINK: Khan Academy Simplifying Square Roots
Exercises Unit 9 Section 2
Multiply the following and simplify where possible.
1.
24 .
6
2.
2 .
4.
5 .
20
5.
( x  6) 3 .
7.
3 .
17
8.
10x 3 . 10 x
13
( x  6)
3.
x3 .
x
6.
3 .
75
9.
3 .
27
Simplify the each square root as much as possible. Show how you split the radicals
into a product for each problem.
10.
8
11.
20
12.
125
13.
63
14.
200
15.
12
16.
48
17.
99
18.
72
19.
98
20.
45
21.
242
22.
56
23.
60
24.
150
25.
a3
26.
x5
27.
z7
28.
ax 2
29.
b2c3
30.
w 4 xz5
31.
4a 3
32.
18b 2
33.
24w 2 x 4
34.
36a
35.
80a 7 b 2 c 3
36.
50u 3 v 5 z
37.
5a 2
38.
68a 2 b 4
39.
49a 5 b
Unit 9 Section 3
Objective

The student will add and subtract square roots with like radicands.
The next operations we need to perform are addition of radicals or square roots. Since
subtraction is the addition of the opposite we can do both addition and subtraction at
the same time. To understand how addition of square roots is done we need to review
how we combine like terms. The examples below should help us remember.
Example A
Example B
x+x+x+x
4x
4a + 3a - a
6a
Example C
2x + 3y + z
cannot be combined
In these examples we see that as long as the variable is the same we can add or
subtract the coefficients of the like terms. We can also see this process when we
use numeric values.
Example A
Example B
1(7) + 1(7) + 1(7) + 1(7)
4(7)
28
4(5) + 3(5) – 1(5)
6(5)
30
In Example A we were adding four sevens together but we could also have been adding
the coefficients and got 4 times 7. In Example B again we added/subtracted the
coefficients to get 6 times 5. It would not really matter what the value in the
parentheses is, the process would be the same. As long as the numbers in the
parentheses are the same we can add the coefficients. The examples below show us
how to add square roots.
Example A
7 +
7
2 7
Example D
4 10 + 5 10
9 10
Example B
5 +
5 +
Example C
3 +
5
3 5
Cannot be added
Example E
7 11 – 12 11 +
-4 11
11
Example F
11
5 3 – 5 3
0
Example G
7 a +
Example H
a
Example I
x 13 – y 13
11 x  3 – 2 x  3
(x – y) 13
9 x3
8 a
VIDEO LINK: Khan Academy Adding and Simplifying Radicals
Exercises Unit 9 Section 3
Multiply the following and simplify where possible.
1.
15 .
4.
8 .
3 .
2 .
2.
5
(a  2) .
5.
12.5
72
(a  2)
3.
x3 .
x5
6.
11 .
11
Simplify each square roots as much as possible. Show how you split the radicals
into a product for each problem.
7.
8.
32
9.
108
44
10.
27
11.
300
12.
112
13.
y3
14.
w3 x 3
15.
9b 7
16.
ab 3
17.
52ab 2 c 3
18.
8w 4
19.
( a  1) 3
20.
150 ab 2 c 4
21.
19w 2 x 4
Answer the following given the expression
a
b
22. What is the ‘a’ called?
23. What is the ‘b’ called?
24. What is the symbol
called?
Add or subtract as indicated. If the radicals can’t be added write N.P. for Not Possible.
25.
2 +
28. 8 a +
2
a
31.
7 +
34.
ab + 4 ab
17
37. 4 5 – 4 5
26. 4 3 – 2 3 + 5 3
27. 7 11 – 13 11
29. 2 13 – 2 13 +
30. a 2 + a 2
32. 14 3 –
35.
13
33. 7 a b – a b
3
( x  4) +
38. 11 ( y  9) –
( x  4)
( y  9)
36. a b + a x
39. 2 x 2 z + 5 x 2 z
Factor the following
40.
y2 + 12y + 32
41. z2 – 4z – 21
42. x2 – 100y2
43.
y2 + 8yz +16z2
44. t2 – 14t + 45
45. w2 + w – 42
46.
5v2 + 16v + 3
47. 4b2 – 2b – 3
48. 6x3 + 12x2 + 6x
Simplify
49. (5xy3)2
50. (2a2)5
51. (w3)-4
Simplify then multiply the resulting monomials. Show your work.
53. (a3b2)4(ab)3
54. (-2a4b)3(-ab3)2
52. (9x4y3z)2
Unit 9 Section 4
Objective

The student will add and subtract square roots.
We know that radicals can only be added or subtracted when the radicals have the same
radicands. There are problems that initially seem to have different radicands. Some of
these problems can be done if we simplify one or both of the radicands. The examples
below show us how to simplify and add radicals.
Example A
20 +
5
The radicands in this example appear to be different. However, the
can be simplified.
20 +
4 .
Since 20 can be factored into 4 times 5 we can
write it as a product of two radicals.
5
5 +
5
2 5 +
After simplifying 20 , the radicands are the
same so we can add the radicals.
5
3 5
Example B
28 +
20
63
The radicands in this example appear to be different. However, both
of the radicals can be simplified.
4 .
28 +
63
7 +
9 .
7
2 7 +3 7
After simplifying, the radicands are the same
so we can add the radicals.
5 7
Example C
125 +
Since 28 is equal to 4 times 7 and 63 has
factors of 9 and 7 so both can be simplified.
45 +
(10)(2)
The radicands in this example appear to be different. However, all
of the radicals can be simplified.
125 +
25 . 5 +
45 –
(10)(2)
9. 5 –
4. 5
5 5 +3 5 –2 5
6 5
Since 125 = 25 . 5, 45 = 9 . 5 and
(10)(2) = 20 = 4 . 5 we can simplify all three.
After simplifying, the radicands are the same
so we can add the radicals.
Example D
200a +
8a 3
The radicands in this example appear to be different. However, all
of the radicals can be simplified.
200a +
100 . 2a +
Since 200a = 100 . 2a and 8a3 = 4a2 . 2a
we can simplify both radicals.
8a 3
4a 2 . 2a
10 2a + 2a 2a
After simplifying, the radicands are the same
so we can add the radicals.
(10  2a) 2a
Exercises Unit 9 Section 4
Multiply the following and simplify where possible.
1.
8 .
4.
10 .
3 .
2.
6
5.
6 .4
2 .
98
(b  9) .
(b  9)
3.
x .
x5
6.
.5 .
50
Simplify each square root as much as possible. Show how you split the radicals
into a product for each problem.
7.
76
8.
10.
y5
11.
13.
ab 4
14.
9.
12
wx 5
12.
16b
50a 3bc 2
15.
27a 2 w 4
250
Answer the following given the expression
a
b
16. What is the radicand?
17. What is the index?
Add or subtract as indicated. If the radicals can’t be added write N.P. for Not Possible.
18.
6 +
19. 7 2 – 5 3
20. 5 10 – 14 10
21. 3 ab + 2 ab
22. -4 13 + 3 13
23. a 5 +
24. x 17 + y 17
25. 5 ( y  1) +
26. 5 a b + 4 a b
6
( y  1)
5
Simplify, then add or subtract as indicated. If the radicals can’t be added write N.P.
for Not Possible.
27.
12 +
27
28.
45 –
20
29.
63 +
28
30.
75 +
108
31.
44 –
176
32.
4a –
144a
33.
32 –
72
34.
8 +
35.
5a 2 +
125a 2
36.
98 +
242
37.
150 –
38.
ab 2 +
a
39. - 7 +
343
40.
10 –
147
24 +
20
6
41. - xy 2 + y x
Solve the following equations by factoring.
42. x2 - 18x + 65 = 0
43. a2 – 5a = 0
44. x2 + 19x + 18 = 0
45. 0 = x2 – 16
46. x2 – 3x = 40
47. a3 = -8a2 – 12
48. 3a2 – 10 = a
49. 8z2 + 4z = 0
50. 6y3 – 9y2 = -3
Divide and list your answer as both a fraction and with negative exponents.
51.
24a 3 x 2 y
6ax 2 y
52.
40ab 4 c 4
2b 2 c
53.
10ab 2 c 4
5abc
Unit 9 Section 5
Objective

The student will divide square roots.
We can now simplify, add, subtract and multiply square roots or radicals. The operation
we still need to study is division. We will use the examples below to investigate how
our method for division will work.
Example A
Simplify
16
4
There are two methods we can use to do this problem.
Method 1
Method 2
16
16
4
4
4
2
4
2
2
Both methods give us the same answer ‘2’. In Method 1 we simplify each
of the two radicals or square roots separately and then divide. In Method 2
we divide the radicands and only then simplify the square root.
Example B
Simplify
400
4
There are two methods we can use to do this problem.
Method 1
Method 2
400
400
4
4
20
2
100
10
10
Both methods give us the same answer ‘10’. In Method 1 we simplify each
of the two radicals or square roots separately and then divide. In Method 2
we divide the radicands and only then simplify the square root.
While both methods work and will produce the same answer Method 2, where we
divide the radicands first, has some advantages. Consider the problem below.
Example C
Simplify
200
8
There are two methods we can use to do this problem.
Method 1
Method 2
200
200
8
8
100  2
25
4 2
10 2
5
2 2
52 2
2 2
5
Both methods give us the same answer ‘50’. In Method 1 we simplify each
of the two radicals, or square roots, separately and then factor and divide. In
Method 2 we divide the radicands and only then simplify the square root.
In order to make our work simpler we should try to divide the radicands whenever
possible. The examples that follow show us how this process can be used.
Example A
Example B
Example C
75
180
245
3
10
5
25
18
49
5
9 .
2
7
3 2
In Example A and C, the division leaves us with a perfect square to simplify while
in Example B, the quotient is not a perfect square but can still be simplified.
Example D
Example E
Example F
a3
15ab 3
72 x 2 yz 5
a
3ab 2
8 yz
a2
9x 2 z 4
5b
3xz2
a
We need to recall
that if we divide
monomials with
exponents we
subtract the powers.
After we divide if we
are not left with
monomials that
simplify then we
are done.
After we divide
sometimes the
monomials will
simplify completely.
VIDEO LINK: Khan Academy Multiply and Simplify a Radical Expression 2
Exercises Unit 9 Section 5
Multiply the following and simplify where possible.
1.
35 .
5 .
7
2.
2 .
xy .
3.
200
x5 y
Simplify each square root as much as possible. Show how you split the radicals
into a product for each problem.
4.
5.
24
6.
175a
14b 3
Add or subtract as indicated. If the radicals can’t be added write N.P. for Not Possible.
7. 4 11 + 9 11
8.
2 – 11 2
9. a 10 + a 10
10. 3 a – 5 a
11. w 13 + 2 13
12.
147 +
27
13.
45 +
500
14.
18 –
128
15.
6b +
54b
16.
98 –
8
17.
18 +
11
18.
12a 3 + a 48a
Divide the radicals and simplify where possible. Show your work.
19.
45
22.
192
5
12
20.
128
23.
275
32
11
21.
216
24.
32
6
2
25.
60
26.
60
29.
5
31.
34.
350
27.
2
3
28.
36
98 x 3
2x
315b 5 c
5b
32.
7
88
30.
11
162
6
120ab
33.
22 xyz
36.
192wx
11z
5a
35.
48ax
8a
4w
Solve the following by completing the square. Round your answers to the nearest
hundredth where needed. Show your work.
37. x2 + 12x + 28 = 0
38. 2x2 + 12x = 8
Use the Quadratic Formula to solve the following. Show your work. Do NOT find
decimal approximations leave square roots in your answers where needed.
39. x2 - 23x + 76 = 0
40. 4a2 – 31a = 8
Unit 9 Section 6
Objective

The student will divide square roots and rationalize denominators.
In all the problems we saw in Section 6 the division did not leave any factors in the
denominator. The problems we will see now could leave radicals in the numerator and
denominator. The best form for expressions with radicals requires that we simplify
the expression and not leave radicals in the denominator. The process for removing
a radical from the denominator is called rationalizing the denominator. There are three
principles we must be able to use in this process. These principles are demonstrated
below.
Any number divided by itself is one.
Example A
Example B
6
a
=1
6
Example C
5 xy
=1
5 xy
a
=1
This is something we have known for a long time. We need to understand that
this principle also applies to square roots or radicals.
Any square root times itself equals the radicand.
Example A
5 .
5 =
25 = 5
Example B
7 .
7 =
Example C
49 = 7
a .
a =
a2 = a
This can be restated as any square root times itself equals the value in the radical.
One times any number equals the original number.
Example A
7 . 1= 7
Example B
5 .1=
Example C
5
5 .
1=
2
5
2
Rationalizing the denominator of a fractional expression requires understanding all
three principles. A given number can be written in many forms. For instance the
value ‘2’ can be written in many ways.
4
, 2.0,
2
2,
2
,
1
4,
8
,…
2
Some of these forms are easier to work with than others but they are all equal to ‘2’.
The preferred form of a fraction with a radical in it is with only one radical in the
numerator. The next set of examples will demonstrate the process of rationalizing the
denominator.
Example A
Rationalize the denominator of
5
3
We cannot change the value of the expression only
the form as we go through this process.
5
3
5 .
3
3
3
When we multiply fractions we multiply straight
across.
5 3
9
The final denominator is the radicand of the square
root in the bottom of the fraction.
5 3
3
Example B
We are multiplying by 1 so the value will not change.
We are using the denominator over itself as our one.
Rationalize the denominator of
6
6
6
6
36
6
6
6
We cannot change the value of the expression only
the form as we go through this process.
1
1 .
6
1
We are multiplying by 1 so the value will not change.
We are using the denominator over itself as our one.
When we multiply fractions we multiply straight
across.
The final denominator is the radicand of the square
root in the bottom of the fraction.
Example C
Rationalize the denominator of
8
2
We cannot change the value of the expression only
the form as we go through this process.
8
2
8 .
2
2
2
The final denominator is the radicand of the square
root in the bottom of the fraction.
8 2
2
42 2
2
We factored the ‘8’ so we could divide out the 2’s.
There will be times when the denominator divides out.
4 2
Example D
We are multiplying by 1 so the value will not change.
We are using the denominator over itself as our one.
Rationalize the denominator of
6
6
6
10 6
6
52 6
3 2
5 6
3
6
We cannot change the value of the expression only
the form as we go through this process.
10
10 .
6
10
We are multiplying by 1 so the value will not change.
We are using the denominator over itself as our one.
The final denominator is the radicand of the square
root in the bottom of the fraction.
We factored the ‘10’ and ‘6’ so we could divide out
the 2’s.
Example E
Rationalize the denominator of
6 a
3b
We cannot change the value of the expression only
the form as we go through this process.
6 a
3b
3b We are multiplying by 1 so the value will not change.
3b We are using the denominator over itself as our one.
6 a .
3b
6 3ab
3b
The final denominator is the radicand of the square
root in the bottom of the fraction.
3  2  3ab
3b
We factored the ‘6’ so we could divide out the 3’s.
2 3ab
b
Fractions inside of radicals can also be simplified with this process. One of the keys to
this process is to change the fraction into a division of square roots. The examples
below illustrate this relationship.
Example F
4
9
Find the
Method 1
Method 2
4
9
4 2
Since 
9 3
4 2
 
9 3
2
we can write the radical as
2
4
9
Because dividing square roots
is done by dividing radicands
we can reverse this and make
a radicand with division into
the dividing of square roots.
2
 
3
The final answer is
9
The final answer is
2
3
2
3
4
This example tells us the best way to simplify the square root of a fraction is to make
it into the division of two square roots. This is Method 2 above.
Example G
16
225
Find the
16
225
16
Write the fraction as the division of square roots.
225
4
25
Example H
We simplified the square roots separately.
25
7
Find the
25
We write the fraction as the division of square roots.
7
We can simplify the numerator but now we need to
rationalize the denominator.
5
7
5 .
7
7
7
The final denominator is the radicand of the square
root in the bottom of the fraction.
5 7
7
Example I
27ab
10b
Find the
27ab
10b
First we divide out the common factor of ‘b’.
27a
We write the fraction as the division of square roots.
10
We can simplify the numerator but now we need to
rationalize the denominator.
3 3a
10
3 3a .
10
We are multiplying by 1 so the value will not change.
We are using the denominator over itself as our one.
10
10
3 30 a
10
We are multiplying by 1 so the value will not change.
We are using the denominator over itself as our one.
The final denominator is the radicand of the square
root in the bottom of the fraction.
We may also be asked to divide two square roots and rationalize the denominator
where necessary.
Example J
Divide and rationalize the denominator of
162
14
We cannot change the value of the expression only
the form as we go through this process.
162
14
81  2
72
81 First we factor the radicands and divide out the
7 common factors.
=
9
We simplified the numerator since it was a perfect square.
7
9 .
7
7
7
9 7
7
VDEO LINK:
We are multiplying by 1 so the value will not change.
We are using the denominator over itself as our one.
The final denominator is the radicand of the square
root in the bottom of the fraction.
Youtube: Rationalizing the Denominator
Exercises Unit 9 Section 6
Simplify each square root as much as possible. Show how you split the radicals
into a product for each problem.
1.
2.
104
3.
117 a
28b 2
Add or subtract as indicated. If the radicals can’t be added write N.P. for Not Possible.
4. 7 17 + 8 17
5.
7. 11 a –
a
8. z 19 –
10.
300
48 +
11.
5 –9 5
20 –
19
125
6.
7 +
9.
80 +
12.
18b +
Divide the radicals and simplify where possible. Show your work.
13.
135
16.
320
5
10
14.
112
17.
75b 2
7
3b
7
15.
88
18.
48a 3
11
2a
5
98b
Divide the radicals and simplify where possible. Rationalize denominators where
necessary. Show your work.
19.
8
20.
14
23.
10
25.
22
26.
31.
10 2b 2
5b
3 2
6
8 3
20
7
24.
10 12
2
11
28.
21.
35
6
22.
7
6
5
27.
15
29.
32.
48ax
15a
8 7
14
1
13
30.
42
35
33.
12
2
Simplify the following. Remember to rationalize the denominators as needed.
34.
36
25
35.
9
16
36.
49
100
37.
10
9
38.
16
5
39.
18
7
40.
a
b
41.
2x
3y
42.
1
4
43.
1
25a 2
44.
1
y
Graph the following quadratic equations using substitution. Show your tables and graphs.
If you completed a square show the work converting the equation’s form.
45. y = (x – 2)2 – 5
46. y = x2 + 6x + 2
Unit 9 Section 7
Objective

The student will solve contextual problems using square roots.
There are many kinds of problems that result in, or use, square roots. Geometric
problems are among the best examples.
Example A
Find an expression for the area of the rectangle below.
3 x +
b
b
A=l.w
First we write the formula for the area.
A = ( b )(3 x +
b)
Second we substitute for ‘l’ and ‘w’.
A = ( b )(3 x +
b)
Now we use the Distributive Property.
A = 3 xb +
Finally we simplify.
b2
A = 3 xb + b
Example B
This is our expression.
Given the rectangle below find the length of the diagonal.
13 m
Geometry Review
To find the length of the diagonal we must use the
Pythagorean Theorem. This theorem states that in
a right triangle the sides have the relationship given
by the diagram and equation below.
x
9 cm
a2 + b2 = c2
92 + 132 = x2
81 + 169 = x2
250 = x2
250 =
25 .
10 = x
5 10 = x
c
a
b
x2
a2 + b2 = c2
Example C
A rectangle has a length of 4 75 cm and a width of 7 12 cm
find the perimeter.
When we have a question about a rectangle often the best way
to start the problem is to draw a rectangle and label the diagram.
4 75
7 12
The perimeter for any figure is found by adding up all the sides.
So for the figure above we should create the equation below.
P = 4 75 + 4 75 + 7 12 + 7 12
We can add some terms.
P = 8 75 + 14 12
Then simplify the radicals.
P= 8.
25 .
P = 8 .5 .
3 + 14 .
4 . 3
3 + 14 . 2 . 3
P = 40 3 + 28 3
P = 68 3
Example D
A rectangle has a length of 3x and a width of x. If the area of
the rectangle is 49 m2 find ‘x’ and the dimensions of the figure.
3x
x
7 .
3
First draw and label the figure.
A=l.w
49 = (3x)(x)
49 = 3x2
3
3
49
= x2
3
We write the formula for the area.
Next we substitute for the variables.
We multiply the monomials.
Use the Division Property of Equality.
49
= x2
3
7
= x
3
Take square roots on both sides.
3
Rationalize the denominator.
3
= x
7 3
= x
3
Simplify the equation.
Simplify the equation.
And we have our answer.
Example E
A rectangle has a length of
x + 3 y and a width of
x –3 y .
Find the expression for the area.
x +3 y
x –3 y
A=l.w
A = ( x + 3 y )( x – 3 y )
We draw and label the rectangle.
We write the formula for the area.
Next we substitute.
The two binomials in the product are actually sum and difference binomials. We
can use either FOIL or our short cut and multiply the binomials.
A = ( x + 3 y )( x – 3 y )
We use the sum and difference multiplication.
A = ( x )2 – (3 y )2
The result of sum & difference multiplication
A=( x .
x ) – (3 y . 3 y ) We expand the squares & multiply the radicals.
A = x2 – 9 y2
A = x – 9y
This is the answer after simplifying the radicals.
In this example there is one step that we need to look at more closely. In the step
starting from A = ( x )2 – (3 y )2 we had to simplify the squares. This can be done
by expanding the squares and multiply the radicals. There is another way to perform
this operation. The examples below will help us understand this method.
Example A
( y )2 =
y .
y =
y2 = y
so
( y )2 = y
Example B
( 5 )2 =
5 .
5 =
25 = 5
so
( 5 )2 = 5
Example B
( ab )2 =
so
( ab )2 = ab
ab .
ab =
a 2 b 2 = ab
From these three examples our conclusion should be:
If we raise a square root to the 2nd power the answer will be the radicand.
Exercises Unit 9 Section 7
Solve the following contextual problems. Show your equations and your work to find
the solutions. All answers must be exact and must be simplified as much as possible.
1. Given the rectangle below as labeled find the area.
24 cm
6 cm
2. Given the right triangle below as labeled find the length of the diagonal.
45 m
5 m
x
3. Given the rectangle below as labeled with an area of 20 in2 find the value of ‘x’ and
the dimensions.
2 x in
8 x in
4. Given the rectangle below as labeled find the area of the figure.
7 + 2 ft
7 – 2 ft
5. Given the rectangle below with the length and diagonal as labeled find the width.
15 cm
19 cm
x
6. Given the circle below has an area of
3
1
 cm2 find the radius. (A =  r 2 )
16
x
7. Given the right triangle as labeled below find the area of the figure.
24 m
10 m
8. Given the rectangle as labeled below find the perimeter.
5 63 cm
3 28 cm
9. Given the square below has an area of 4
21 2
in find the side. (A = s2)
25
x
10. If a triangle has sides of 2 45 cm,
80 cm, and 3 20 cm find the perimeter.
11. If a rectangle has length of 8x, a width of 2x, and an area of 81 m2, find
the value of ‘x’ and the dimensions.
12. If a rectangle has sides of a +
5a and
3a find the expression for the area.
13. Given a right triangle as shown below find the hypotenuse.
x
5 m
2 5m
14. If a rectangle has a width of
and the perimeter.
15. If a circle has a radius of
2 cm and a diagonal of 16 cm find the length
7a find the expression for its area.
16. Given the trapezoid below as labeled find the area of the figure. (A =
1
h (b1  b2 ) )
2
7 5 in
20 in
5 5 in
17. If a square has a side of 2x and an area of 169 cm2 find the value of ‘x’ and
the length of the side.
18. Mike has built a backdrop panel for a school play. The length of the panel is
80 ft. and the height is 45 ft. He needs to put a diagonal stabilizer on the
panel. What will the measurement of the diagonal need to be?
19. A solar panel needs to have an area of 256 in2. The panel will be a rectangle.
The length should be nine times its width. Find the dimensions of the panel.
20. Given the isosceles triangle below as labeled find the altitude of the triangle.
x
2 71 ”
200 ”
Unit 9 Section 8
Objective

The student will solve literal equations using square roots.
Solving literal equations means that we are solving an equation for a particular variable.
This is a skill we learned in Unit 2. Solving a literal equation will result in the isolated
variable being equal to an expression not just a real number. The two examples below
illustrate this difference.
Example A
x2 − 7x + 12 = 0
(x − 3)(x − 4) = 0
x = 3, 4
Example B
ab + c = d solve for b
-c -c
ab = d − c
a
a
b=d−c
a
In Example A we solved the equation and the variable 'x' was equal to 3 or 4, real
numbers. In Example B we isolated the variable 'b' however the result is an expression.
Example B shows us the process for solving a literal equation.
When we studied this skill in Unit 2 we used it to isolate variables that had the
understood exponent of 1. We have progressed beyond this point and we can now
have quadratic equations or even equations with radicals. The examples that follow
show us how to isolate variables from quadratic and radical equations.
Example A
Given the formula
A = 6s2 isolate s. (The surface area of a cube)
A  6s 2
A 6s 2

6
6
A
 s2
6
A
s
6
A. 6
s
6
6
6A
s
6
We use the Division Property of Equality.
We take the square root on both sides in
order to reverse the square of the 's2' term.
We need to rationalize the denominator.
This is the final answer.
Given y = (x - b)2 - c
Example B
isolate x.
y  ( x  b) 2  c
y  ( x  b) 2  c
c
c
The square is now isolated.
y  c  ( x  b) 2
y  c  ( x  b) 2
y c  xb
b
b
We take square roots on both sides. When the
square is isolated the next step is taking the square
root.
We use the Addition Property of Equality.
y c b  x
Example C
We use the Addition Property of Equality.
Given
w  2 z 1
isolate z.
w  2 z 1
We use the Division Property of Equality.
w  2 z 1
2
2
w

2
z 1
2
 w
  
2

z 1
w2
 z 1
4
1
1
w2
1  z
4

2
Since the square root is now isolated we can
reverse the root and square both sides.
We use the Subtraction Property of Equality.
Exercises Unit 9 Section 8
Isolate the asked for variable in each formula or equation. Show all your work and be
prepared to explain all your steps. Rationalize all denominators.
1. A = s2
3. a2 + b2 = c2
5. V =
2. A =  r 2
isolate s
1 2
r h
3
isolate r
isolate c
4. a2 + b2 = c2
isolate a
isolate r
6. w = z2 + 1
isolate z
7. z = (a + 2)2 − 1
8. z = 3y2
isolate a
isolate y
9. 4m = 2(w − z)2 isolate w
10.
y
11.
12.
w 2 z  5
a
2b  1
isolate b
xz
isolate x
isolate z
Find the square roots. If there is no real answer then write “no real number”.
13.
9
14.
49
15. - 100
Estimate the square roots to the nearest tenth. Check your answers by multiplying
your estimate times itself. Show your work.
16.
17.
23
139
Evaluate the following. Show your work.
If the answer is non-real state "no real answer".
x4 –1
18. Evaluate y =
with x = 117
19. Evaluate y = 5 2 x  3 +
20. Evaluate y =
1
3
x 2  19
21. Evaluate y = - 4 x  5
5x  4
22. Given
f(x) =
23. Given
g(x) = 2 x  5
19
with x = 11
with x = 10
with x = 44
Find f(9), f(1), and f(10) show your work.
Find g(9), g(30), and g(1) show your work.
Multiply the following and simplify where possible.
24.
51 .
3
25.
27.
5 .
11
28.
30.
abc 3 .
a 3b 3 c
31.
5 .
26.
z .
( x  4)
29.
2a .
72a
10a
32.
ab .
w
20
( x  4) .
5a .
2 .
z
Simplify the each square root as much as possible. Show how you split the radicals
into a product for each problem.
33.
153
34.
360
35.
18x 3
36.
ab 4
37.
50a 3bc 2
38.
27a 2 w 4
Add or subtract as indicated. If the radicals can’t be added write N.P. for Not Possible.
39.
11 +
42. x 10 +
11
10
40. 7 5 – 7 3
43.
41. 5 7 – 18 7
( y  1) + 7 ( y  1)
44. 3b a – 4b a
Simplify, then add or subtract as indicated. If the radicals can’t be added write N.P.
for Not Possible.
45.
10 +
48.
6 +
90
14
46.
18 –
50
49.
7 –
28 +
112
47.
24 +
54
50.
2a +
128a
Divide the radicals and simplify where possible. Show your work.
51.
125
54.
60b 5
5
5b
52.
128
55.
56ax 2
2
8a
53.
63wx
56.
27 w
7x
3w
Divide the radicals and simplify where possible. Rationalize denominators where
necessary. Show your work.
57.
15
58.
33
59.
35
6
60.
5
61.
14
28 20
14
62.
10 3z 2
2z
21
11
Simplify the following. Remember to rationalize the denominators as needed.
63.
64
25
64.
49
16
65.
7
9
66.
1
4a 2
67.
2
z
68.
1
5