Name______________________________
Date_______________
Integrated Algebra A
Notes/Homework Packet 5
Lesson
Introduction to Square Roots
Simplifying Radicals
Simplifying Radicals with Coefficients
Adding & Subtracting Radicals
Adding & Subtracting Radicals continued
Multiplying Radicals
Dividing Radicals
Pythagorean Theorem Introduction
Pythagorean Theorem Word Problems
Review Sheet
Test #5
Homework
HW #1
HW #2
HW #3
HW #4
HW #5
HW #6
HW #7
HW #8
HW #9
1
Introduction to Square Roots
Taking the square root of a number is the opposite of squaring the number. Even
your calculator knows this because x2 has
above it. To find a square root, hit 2nd
button , select
, put the number in, close the parentheses and hit enter!
Every positive number has two square roots: one positive and one negative.
For example:
25 = 5 and
25 = -5
because 52 = 25 and (-5)2 = 25
Let’s practice – These are the ones we should know for this unit! But of course, there
are more than just these ones!
12 =
(-1)2 =
1=
22 =
(-2)2 =
4 =
32 =
(-3)2 =
9 =
42 =
(-4)2 =
16 =
52 =
(-5)2 =
25 =
62 =
(-6)2 =
36 =
72 =
(-7)2 =
49 =
82 =
(-8)2 =
64 =
92 =
(-9)2 =
81 =
102 =
(-10)2 =
100 =
112 =
(-11)2 =
121 =
122 =
(-12)2 =
144 =
Now that we have these perfect squares, we can combine them and do some
operations! When we do these operations, we only use the positive value of the
square root.
2
Example 1:
Example 3:
Example 5:
18  2
49  100
2 16  5
Example 2:
Example 4:
Example 6:
3(4)  13
144
9
 81  10
Practice
1.
36  121
2.
3.
100
25
4.
3 4  2 64
21  5  7 5(5)
3
Name________________________________
Date_________________
HW #1
1. Find the two square roots of the following numbers (one positive, one negative):
a) 64
b) 100
c) 16
d) 225
2. Evaluate each expression:
a)
c)
e)
16  7
10( 100  5)
2 36
6
b)
9  16
d)
25 * 49
f)
2 100  75
Review
1) Solve and show all work
42 – [2(8+3)-4]2
4
Simplifying Radicals
When simplifying radicals, you must know the perfect squares.
VIPS : Very Important Perfect Squares
VIPS:
Examples:
22 =
STEPS:
1) Find the largest perfect square that
divides evenly into the number inside
the radical. Put him under the first
.
32 =
2) Put “his friend” in the 2nd
42 =
3) Take the
52 =
leave the 2nd # in the
62 =
4) Make sure your final
is totally
reduced!! If not, repeat process.
12 =
72 =
1) Simplify 18
2) Simplify
.
the first number and
.
24
82 =
92 =
102 =
112 =
122 =
3) Simplify 160
132 =
142 =
152 =
162 =
4) Simplify 17
5
Simplify these radicals:
1)
20
2)
24
3)
40
4)
10
5)
50
6)
300
2)
27
3)
48
6)
63
Practice
1)
4)

28
5)
90

98

6
Name________________________________
Date_________________
HW #2
Simplify the following radicals, showing ALL WORK:
1)
75
2)
45
3)
32
4)
96
5)
54
6)
200
7)
117
8)
98
9)
108
Review:
1) Create a stem and leaf plot of the following data set.
11, 21, 3, 35, 22, 19, 8, 37, 42, 13, 4
7
Simplify the following radicals:
1. 125
2.
54
3.
80
Simplifying Radicals with Coefficients
When we put a coefficient in front of the radical, we are multiplying it by our answer
after we simplify.
If we take Warm up question #1 and put a 6 in front of it, it looks like this
6 125
6  25
5
We keep bringing
down each piece
and multiply at the
end.
6 5 5
30 5
1. 2 18
2. -4 12
3. 6 72
8
Examples
1
20
2
2. 10 32
3. -2 48
5. 3 13
6. 5 500
7. 3 250
8. -5 24
9.
10. 3 27
11. - 45
12. 12 60
1.
4. - 44
Practice
4
50
5
9
Name________________________________
Simplify the following radicals:
1. 28
2.
600
1
162
3
4. 5 8
5.
7. 10 80
8. - 99
Date________________
HW #3
3.
98
6. -7 45
9. 3 15
Review:
1) The length and width of a rectangle are in the ratio 3:4. The perimeter of the
rectangle is 84 cm. Find the length and width of the rectangle.
10
Adding / Subtracting Radicals
1) Simplify
50
2) Simplify
90
Important Points to know:
 Make sure the radicals are in ____________ _______ before you add or subtract.
 In order to add or subtract radicals, the number inside the radicals must be the
________. This is called the ______________.
 When the radicands are the same, then, you can add or subtract only the
numbers in __________ of the radicals (_________________). The radicands are
treated kind of like variables.
Already-Simplified Radicals:
Example 1:
Example 2:
2 +
2
x+x
1 2 +1 2
1x + 1x
= 2 2
= 2x
2 3 + 4 3
Example 3:
NOTE:
-These numbers can be
“added” because the
radicands are the same.
-However, only the numbers
in front, which are 1’s, are
added. Nothing happens to
the 2 . It is almost like an
x.
6 5 – 4 5
Practice
1)
7 6 +2 6
2)
13 + 5 13
3)
4 11 – 11
4)
2 3 –6 3
5)
-10 2 + 3 2
6)
-8 15 – 9 15
11
Un-Simplified Radicals:
When the radicals are NOT in simplified form, we must use the method learned the
last couple of days to simplify them!
3 +
Example 4:
NOTE:
The 3 is simplified already, but the
27
27 must still be simplified.
3 +
9
3
3 + 3 3
=
Example 5:
4 2 + 3 50
Example 6:
3 20 – 2 5
Practice
1)
2 3 + 4 12
2)
3 5 – 2 45
3)
7 5 – 15
4) Find the perimeter of a rectangle whose length is 3 5 and width is 2 7 . [Draw a
picture!]
12
Name________________________________
Date_________________
HW #4
Perform the indicated operation (Add or Subtract):
1)
3 +8 3
2)
3 5 – 7 5
4) The sum of 12 and 5 3 is?
6) Simplify:
8)
5 3 +
200 – 3 2
27
3)
11 – 11
5) Find the difference of 12 11 and
44 .
7) Express the sum of 18 + 5 2 in simplest
radical form.
9)
5 3 +2 3 –6 3
10) Find the perimeter of a rectangle whose length is 4 5 and width is 3 7 .
[Draw a picture!]
13
Adding/Subtracting Radicals continued
1)
2) 18  24
3 2 3
Sometimes we need to simplify more that one radical in order to be able to add or
subtract them.
Example 1:
Example 2:
18  32
9 2
+
3 2
+
We need to simplify
both terms to see if
we have the same
radicands!!!
16 2
4 2
48  27
We have the same radicands so we can perform addition!
Example 3:
2 80  45
Let’s do some example that might not have the same radicands in the end.
Example 4:
32  54
Example 5:
72  3 20
14
More Examples:
1.
12  108
2.
 24  96
3.
2 8  27
Practice:
Simplify the following expressions.
1.
9 50
2.
28  63
4.
7  175
5.
1
40
2
7.
27  32
8.
4 22
3.
6.
4 14  6 14
80  20
15
Name______________________________
1.
4.
7.
10.
18  50
11 45
5  125
32  75
2.
5.
8.
Date_____________
HW #5
 80  45
50  98
1
32
4
11.
3.
6.
9.
 8  32
9 7 6 7
24  54
 8 13
12. Find the perimeter of a rectangle whose length is 3 10 and width is 4 2 .
[Draw a picture!]
16
Multiplying Radicals
9  9=________
6  6=_______
9  9 =________=_____
10  10=_________
6  6 =________=____
10  10 =________=____
Notice how when we multiply the same square root by itself, the answer becomes
the radicand (WITHOUT THE RADICAL SIGN)!
How about this…
 7
 5
2
=
7  7 = _________=_____
2
= ____

_____= _________=_____
The square root symbol
and the exponent
2
____________ each other out and
leave the ______________________ as our __________________.
*When we add or subtract radicals they must have the same radicand. This is NOT
necessarily true for multiplying (and dividing)!
Example 1:
12  3 = ______________=________
Example 2:
2  32
= _____________ =________
Example 3:
25  4
= ______________=________
Example 4:
18  8
= _____________ =________
17
*Sometimes when we multiply we do not get a perfect square. In that case, we must
simplify our answer!
Example 1:
6  2 = __________
Example 2:
12  6 =__________
Example 3:
15  3 = _________
Example 4:
2  22 =__________
*One more thing we must deal with when multiplying radicals is coefficients!
Step 1: We must multiply the
coefficients (outsides)
2 5 3 8
Step 2: We must multiply the
radicals (insides)
23 58
Step 3: Simplify if necessary!
6 40
6 40
Coefficient: The
number in FRONT of
the radical.
Now let’s simplify
Practice:
1.
5 6 4 8
2.
 4 3  7 15
3.
2 5  4 10
18
Name_________________________________
Date_________________
HW #6
Multiply the radicals. Make sure to reduce all answers into simplest form!
1.
4.
7.
4 4
3 2  9 20
18  2
2.
 3
5.
 12  6 5
6.
8.
2 16  3 4
9.
2
Review: Perform the given operation
1. 3 75  2 27
2. 6 18  4 32
3.
10  8
 16 
2
26  26
3.  12  4 8
19
1)
4 6 9 3
2) 9 6 
2
15
3

Dividing Radicals
*When dividing radicals, we follow the same procedure as multiplying radicals. Now
we divide the coefficients (outsides) and divide the radicals (insides).
*Sometimes when dividing radicals you get a whole number, which makes simplifying
easy!
Example 1:
72
=
8
9 = 3
Example 2:
50
=
Example 4:
2
Example 3:
3
=
3
2
48
Here, we can just DIVIDE 72 by 8 and
make a new radical with that answer.
Then, simplify the radical if possible.
=
Remember that anything
divided by itself is 1
(they cancel each other out).
Example 5:
96
3
=
*When there are numbers in front of the radicals (coefficients) you must divide those
too! Be sure to leave coefficients in fraction form.
20
Example 6:
6 10
=
3 2
Example 7:
3 54
=
6 3
*What if we take the radical of a fraction?
Example 1:
4
=
9
4
9
=
2
3
First, take the square
root of the
numerator; then,
take the square root
of the denominator,
SEPARATELY!!!
Example 2:
16
=
25
Practice: Divide; then simplify the quotient.
1)
4)
7)
9
49
40
5
2)
5)
9 6
3 6
25 24
5 2
3)
6)
20
5
12 60
6 5
3 120
9 5
21
Name_______________________________
Date__________________
HW #7
Divide; then simplify the quotient.
1)
4)
7)
36
9
300
5
25
36
2)
5)
8)
8 2
2 2
2 33
11
35 108
7 4
3)
6)
9)
150
3
8 48
2 3
9
64
Review
Write an algebraic expression or equation.
1) Five times the sum of 3 and a number.
2) The sum of 7 and a number exceeds a 3 times a number by 5.
22
Pythagorean Theorem
In any RIGHT triangle,
the sum of the squares
of the lengths of the two
legs is equal to the
square of the length of
the hypotenuse.
Hypotenuse
c
a
a2 + b2 = c2
b
Legs
Example 1: Find the length of the hypotenuse.
x
4
3
Example 2: Find the length of the hypotenuse.
Example 3: Find the missing side of the triangle.
13
5
x
23
Example 4: Find the missing side in simplest radical form.
x
14cm
10cm
Example 5:
13
Find the unknown leg in the right triangle, in simplest
radical form.
3
x
Practice: Find the length of the missing side. Keep answer in simplest
radical form.
1.
2.
x
20in
7
8
15ft
25ft
3.
15in
x
x
24
Name_______________________________
Date__________________
HW #8
1. Find the length of the hypotenuse of this right triangle. Round to the nearest
tenth.
x
5
2
2. Find the length of the hypotenuse of this right triangle.
15m
8m
x
3. Solve for the unknown side in this right triangle.
5
x
13
4. Solve for the unknown side in this right triangle. Put your answer in simplest radical
form.
x
6in
12in
25
5. Solve for the unknown side in this right triangle. Round to the nearest
thousandth.
10
x
14
6. Solve for the unknown side in this right triangle. Put your answer in simplest radical
form.
9
x
8
Review:
1.
Solve for x: 18x – (4x – 10) = 24
2.
Check your answer for Review #1.
3.
After a 5-inch-by-7-inch photograph is enlarged, its shorter side measures 20
inches. Find the length in inches of its longer side. [Draw Pictures!!!]
26
Pythagorean Theorem Word Problems
Solve for x.
1.
2.
x
x
10ft
8cm
20ft
6cm
Word Problems with the Pythagorean Theorem:
Steps:
 Read the problem.
 Identify key elements.
 Draw a picture.
 Solve for the missing side.
 Label your answer!
1. A ramp was constructed to load a truck. If the ramp is 9 feet long and the
horizontal distance from the bottom of the ramp to the truck is 7 feet, what is the
vertical height of the ramp?
27
2. There is a 13-foot ladder leaning against the side of a building. The ladder reaches
up the building 12 feet. How far is the bottom of the ladder from the bottom of the
building?
3. Find the diagonal of a square whose sides are 5cm long.
4. Ms. Green tells you that a right triangle has a hypotenuse of 13 and a leg of 5. She
asks you to find the other leg of the triangle. What is your answer?
5. A suitcase measures 24 inches long and 18 inches high. What is the diagonal
length of the suitcase to the nearest tenth of a foot? [Note: Once you find your
answers in inches, you must convert it to feet!]
28
Name: _______________________________
Date: _______________
HW #9
1. A wall is supported by a brace 10 feet long, as shown in the diagram below. If
one end of the brace is placed 6 feet from the base of the wall, how many feet up
the wall does the brace reach?
10ft
6ft
2. The two legs of a right triangle are 9 and 7. Find the hypotenuse of the triangle.
Draw a picture! Leave your answer in radical form.
3. How many feet from the base of a house must a 39-foot ladder be placed so that
the top of the ladder will reach a point on the house 36 feet from the ground? Draw
a picture!
Review
Find the perimeter of the triangle below. Show all work for final answer!
*Hint: Need to find the missing side first.
30cm
50cm
29