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Name______________________________ Date_______________

Integrated Algebra A

Notes/Homework Packet 5

|Lesson |Homework |

|Introduction to Square Roots |HW #1 |

|Simplifying Radicals |HW #2 |

|Simplifying Radicals with Coefficients |HW #3 |

|Adding & Subtracting Radicals |HW #4 |

|Adding & Subtracting Radicals continued |HW #5 |

|Multiplying Radicals |HW #6 |

|Dividing Radicals |HW #7 |

|Pythagorean Theorem Introduction |HW #8 |

|Pythagorean Theorem Word Problems |HW #9 |

|Review Sheet | |

|Test #5 | |

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 [pic]above it. To find a square root, hit 2nd button , select [pic] , put the number in, close the parentheses and hit enter!

Every positive number has two square roots: one positive and one negative.

For example:

[pic] = 5 and [pic] = -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 = [pic]=

22 = (-2)2 = [pic] =

32 = (-3)2 = [pic] =

42 = (-4)2 = [pic] =

52 = (-5)2 = [pic] =

62 = (-6)2 = [pic] =

72 = (-7)2 = [pic] =

82 = (-8)2 = [pic] =

92 = (-9)2 = [pic] =

102 = (-10)2 = [pic] =

112 = (-11)2 = [pic] =

122 = (-12)2 = [pic] =

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.

Example 1: [pic] Example 2: [pic]

Example 3: [pic] Example 4: [pic]

Example 5: [pic] Example 6: [pic]

Practice

1. [pic] 2. [pic]

3. [pic] 4. [pic]

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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) [pic] b) [pic]

c) [pic] d) [pic]

e) [pic] f) [pic]

Review

1) Solve and show all work

42 – [2(8+3)-4]2

Simplifying Radicals

When simplifying radicals, you must know the perfect squares.

VIPS : Very Important Perfect Squares

Examples:

1) Simplify [pic]

2) Simplify [pic]

3) Simplify [pic]

4) Simplify [pic]

Simplify these radicals:

1) [pic] 2) [pic] 3) [pic]

4) [pic] 5) [pic] 6) [pic]

Practice

1) [pic] 2) [pic] 3) [pic]

4) [pic] 5) [pic] 6) [pic]

Name________________________________ Date_________________

HW #2

Simplify the following radicals, showing ALL WORK:

1) [pic] 2) [pic] 3) [pic]

4) [pic] 5) [pic] 6) [pic]

7) [pic] 8) [pic] 9) [pic]

Review:

Create a stem and leaf plot of the following data set.

11, 21, 3, 35, 22, 19, 8, 37, 42, 13, 4

Simplify the following radicals:

1. [pic] 2. [pic] 3. [pic]

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[pic]

6[pic][pic][pic]

6[pic]5[pic]

30[pic]

1. 2[pic] 2. -4[pic] 3. 6[pic]

Examples

1. [pic] 2. 10[pic] 3. -2[pic]

4. -[pic] 5. 3[pic] 6. 5[pic]

Practice

7. 3[pic] 8. -5[pic] 9. [pic]

10. 3[pic] 11. -[pic] 12. 12[pic]

Name________________________________ Date________________

HW #3

Simplify the following radicals:

1. [pic] 2. [pic] 3. [pic]

4. 5[pic] 5. [pic] 6. -7[pic]

7. 10[pic] 8. -[pic] 9. 3[pic]

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.

Adding / Subtracting Radicals

1) Simplify [pic] 2) Simplify [pic]

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: [pic] + [pic] x + x

1[pic] + 1[pic] 1x + 1x

= 2[pic] = 2x

Example 2: 2[pic] + 4[pic] Example 3: 6[pic] – 4[pic]

Practice

1) 7[pic] + 2[pic] 2) [pic] + 5[pic] 3) 4[pic] – [pic]

4) 2[pic] – 6[pic] 5) -10[pic] + 3[pic] 6) -8[pic] – 9[pic]

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!

Example 4: [pic] + [pic]

[pic] + [pic][pic]

[pic] + 3[pic]

=

Example 5: 4[pic] + 3[pic] Example 6: 3[pic] – 2[pic]

Practice

1) 2[pic] + 4[pic] 2) 3[pic] – 2[pic] 3) 7[pic] – [pic]

4) Find the perimeter of a rectangle whose length is 3[pic] and width is 2[pic]. [Draw a

picture!]

Name________________________________ Date_________________

HW #4

Perform the indicated operation (Add or Subtract):

1) [pic] + 8[pic] 2) 3[pic] – 7[pic] 3) [pic] – [pic]

4) The sum of [pic] and 5[pic] is? 5) Find the difference of 12[pic] and [pic].

6) Simplify: [pic] – 3[pic] 7) Express the sum of [pic] + 5[pic] in simplest

radical form.

8) 5[pic] + [pic] 9) 5[pic] + 2[pic] – 6[pic]

10) Find the perimeter of a rectangle whose length is 4[pic] and width is 3[pic].

[Draw a picture!]

Adding/Subtracting Radicals continued

1) [pic] 2) [pic]

Sometimes we need to simplify more that one radical in order to be able to add or subtract them.

Example 1: [pic]

[pic] + [pic]

3[pic] + 4[pic] We have the same radicands so we can perform addition!

Example 2: [pic] Example 3: [pic]

Let’s do some example that might not have the same radicands in the end.

Example 4: [pic] Example 5: [pic]

More Examples:

1. [pic] 2. [pic] 3. [pic]

Practice:

Simplify the following expressions.

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic]

Name______________________________ Date_____________

HW #5

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

10. [pic] 11. [pic]

12. Find the perimeter of a rectangle whose length is 3[pic] and width is 4[pic].

[Draw a picture!]

Multiplying Radicals

9[pic]9=________ 6[pic]6=_______ 10[pic]10=_________

[pic]=________=_____ [pic]=________=____ [pic]=________=____

Notice how when we multiply the same square root by itself, the answer becomes the radicand (WITHOUT THE RADICAL SIGN)!

How about this…

[pic] = [pic]= _________=_____

[pic] = ____ [pic] _____= _________=_____

The square root symbol [pic]and the exponent [pic] ____________ 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: [pic]= ______________=________

Example 2: [pic] = _____________ =________

Example 3: [pic]= ______________=________

Example 4: [pic] = _____________ =________

*Sometimes when we multiply we do not get a perfect square. In that case, we must simplify our answer!

Example 1: [pic]= __________ Example 2: [pic]=__________

Example 3:[pic]= _________ Example 4: [pic]=__________

*One more thing we must deal with when multiplying radicals is coefficients!

[pic]

[pic]

[pic] [pic]

[pic] Now let’s simplify

Practice:

1. [pic] 2. [pic] 3. [pic]

Name_________________________________ Date_________________

HW #6

Multiply the radicals. Make sure to reduce all answers into simplest form!

1. [pic][pic][pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic][pic]

Review: Perform the given operation

1. [pic] 2. [pic] 3. [pic]

1) [pic] 2) 9[pic] [pic] [pic]

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:

[pic] = [pic] = 3

Example 2: Example 3:

[pic] = [pic] =

Example 4: Example 5:

[pic] = [pic] =

*When there are numbers in front of the radicals (coefficients) you must divide those too! Be sure to leave coefficients in fraction form.

Example 6: Example 7:

[pic] = [pic] =

*What if we take the radical of a fraction?

Example 1: Example 2:

[pic] = [pic] = [pic] [pic] =

Practice: Divide; then simplify the quotient.

1) [pic] 2) [pic] 3) [pic]

4) [pic] 5) [pic] 6) [pic]

7) [pic]

Name_______________________________ Date__________________

HW #7

Divide; then simplify the quotient.

1) [pic] 2) [pic] 3) [pic]

4) [pic] 5) [pic] 6) [pic]

7) [pic] 8) [pic] 9) [pic]

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.

Pythagorean Theorem

Example 1: Find the length of the hypotenuse.

Example 2: Find the length of the hypotenuse.

Example 3: Find the missing side of the triangle.

Example 4: Find the missing side in simplest radical form.

Example 5: Find the unknown leg in the right triangle, in simplest radical form.

Practice: Find the length of the missing side. Keep answer in simplest radical form.

1. 2. 3.

Name_______________________________ Date__________________

HW #8

1. Find the length of the hypotenuse of this right triangle. Round to the nearest

tenth.

2. Find the length of the hypotenuse of this right triangle.

3. Solve for the unknown side in this right triangle.

4. Solve for the unknown side in this right triangle. Put your answer in simplest radical form.

5. Solve for the unknown side in this right triangle. Round to the nearest

thousandth.

6. Solve for the unknown side in this right triangle. Put your answer in simplest radical

form.

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!!!]

Pythagorean Theorem Word Problems

Solve for x.

1. 2.

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?

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!]

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?

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.

-----------------------

15ft

25ft

x

15in

x

13

5

x

13

3

x

x

4

14cm

20in

10cm

Remember that anything divided by itself is 1

(they cancel each other out).

Here, we can just DIVIDE 72 by 8 and make a new radical with that answer. Then, simplify the radical if possible.

First, take the square root of the numerator; then, take the square root of the denominator, SEPARATELY!!!

Step 1: We must multiply the coefficients (outsides)

Step 2: We must multiply the radicals (insides)

Step 3: Simplify if necessary!

Coefficient: The number in FRONT of the radical.

We need to simplify both terms to see if we have the same radicands!!!

NOTE:

The [pic] is simplified already, but the [pic]must still be simplified.

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.

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 [pic]. It is almost like an x.

We keep bringing down each piece and multiply at the end.

VIPS:

12 =

22 =

32 =

42 =

52 =

62 =

72 =

82 =

92 =

102 =

112 =

122 =

132 =

142 =

152 =

162 =

STEPS:

1) Find the largest perfect square that divides evenly into the number inside the radical. Put him under the first [pic].

2) Put “his friend” in the 2nd [pic].

3) Take the [pic]the first number and leave the 2nd # in the [pic].

4) Make sure your final [pic] is totally reduced!! If not, repeat process.

3

a2 + b2 = c2

Hypotenuse

Legs

c

b

a

x

7

8

x

20ft

x

10ft

6cm

8cm

x

x

5

2

15m

8m

x

5

x

13

x

12in

6in

10

x

14

8

9

x

10ft

6ft

30cm

50cm

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