CH EN 3453 Heat Transfer 2014 Fall Utah Homework HW 08 Solutions
Name______________________________ · Web viewNotes/Homework Packet 5. Lesson Homework Introduction...
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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 1
Transcript of Name______________________________ · Web viewNotes/Homework Packet 5. Lesson Homework Introduction...
Name______________________________ Date_______________
Integrated Algebra ANotes/Homework Packet 5
Lesson HomeworkIntroduction to Square Roots HW #1
Simplifying Radicals HW #2Simplifying Radicals with Coefficients HW #3
Adding & Subtracting Radicals HW #4Adding & Subtracting Radicals continued HW #5
Multiplying Radicals HW #6Dividing Radicals HW #7
Pythagorean Theorem Introduction HW #8Pythagorean Theorem Word Problems HW #9
Review SheetTest #5
1
Introduction to Square RootsTaking 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:
= 5 and = -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 = =
22 = (-2)2 = =
32 = (-3)2 = =
42 = (-4)2 = =
52 = (-5)2 = =
62 = (-6)2 = =
72 = (-7)2 = =
82 = (-8)2 = =
92 = (-9)2 = =
102 = (-10)2 = =
112 = (-11)2 = =
122 = (-12)2 = =
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: Example 2:
2
Example 3: Example 4:
Example 5: Example 6:
Practice
1. 2.
3. 4.
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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) b)
c) d)
e) f)
Review1) Solve and show all work
42 – [2(8+3)-4]2
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Simplifying Radicals
When simplifying radicals, you must know the perfect squares.
VIPS : Very Important Perfect SquaresExamples:
1) Simplify
2) Simplify
3) Simplify
4) Simplify
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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 .
2) Put “his friend” in the 2nd .
3) Take the the first number and leave the 2nd # in the .
4) Make sure your final is totally reduced!! If not, repeat process.
Simplify these radicals:
1) 2) 3)
4) 5) 6)
Practice
1) 2) 3)
4) 5) 6)
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Name________________________________ Date_________________HW #2
Simplify the following radicals, showing ALL WORK:
1) 2) 3)
4) 5) 6)
7) 8) 9)
Review:1) Create a stem and leaf plot of the following data set.
11, 21, 3, 35, 22, 19, 8, 37, 42, 13, 4
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Simplify the following radicals:1. 2. 3.
Simplifying Radicals with CoefficientsWhen 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
6
6 5
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1. 2 2. -4 3. 6
8
We keep bringing down each piece and multiply at the
end.
Examples
1. 2. 10 3. -2
4. - 5. 3 6. 5
Practice
7. 3 8. -5 9.
10. 3 11. - 12. 12
Name________________________________ Date________________HW #3
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Simplify the following radicals:1. 2. 3.
4. 5 5. 6. -7
7. 10 8. - 9. 3
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.
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Adding / Subtracting Radicals1) Simplify 2) Simplify
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: + x + x
1 + 1 1x + 1x
= 2 = 2x
Example 2: 2 + 4 Example 3: 6 – 4
Practice
1) 7 + 2 2) + 5 3) 4 –
4) 2 – 6 5) -10 + 3 6) -8 – 9
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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 . It is almost like an x.
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: +
+
+ 3
=
Example 5: 4 + 3 Example 6: 3 – 2
Practice
1) 2 + 4 2) 3 – 2 3) 7 –
4) Find the perimeter of a rectangle whose length is 3 and width is 2 . [Draw a picture!]
Name________________________________ Date_________________HW #4
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NOTE:The is simplified already, but the must still be simplified.
Perform the indicated operation (Add or Subtract):
1) + 8 2) 3 – 7 3) –
4) The sum of and 5 is? 5) Find the difference of 12 and .
6) Simplify: – 3 7) Express the sum of + 5 in simplest radical form.
8) 5 + 9) 5 + 2 – 6
10) Find the perimeter of a rectangle whose length is 4 and width is 3 . [Draw a picture!]
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Adding/Subtracting Radicals continued
1) 2)
Sometimes we need to simplify more that one radical in order to be able to add or subtract them.
Example 1:
+
3 + 4 We have the same radicands so we can perform addition!
Example 2: Example 3:
Let’s do some example that might not have the same radicands in the end.
Example 4: Example 5:
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We need to simplify both terms to see if we have the same
More Examples:
1. 2. 3.
Practice:Simplify the following expressions.
1. 2. 3.
4. 5. 6.
7. 8.
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Name______________________________ Date_____________HW #5
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11.
12. Find the perimeter of a rectangle whose length is 3 and width is 4 . [Draw a picture!]
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Multiplying Radicals
9 9=________ 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…
= = _________=_____
= ____ _____= _________=_____
The square root symbol and the exponent ____________ 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: = ______________=________
Example 2: = _____________ =________
Example 3: = ______________=________
Example 4: = _____________ =________
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*Sometimes when we multiply we do not get a perfect square. In that case, we must simplify our answer!
Example 1: = __________ Example 2: =__________
Example 3: = _________ Example 4: =__________
*One more thing we must deal with when multiplying radicals is coefficients!
Now let’s simplify
Practice:1. 2. 3.
Name_________________________________ Date_________________HW #6
Multiply the radicals. Make sure to reduce all answers into simplest form!
1. 2. 3.
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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.
4. 5. 6.
7. 8. 9.
Review: Perform the given operation1. 2. 3.
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1) 2) 9
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:
= = 3
Example 2: Example 3:
= =
Example 4: Example 5:
= =
*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: = =
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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).
First, take the square root of the numerator; then, take the square root of the denominator, SEPARATELY!!!
*What if we take the radical of a fraction?
Example 1: Example 2:
= = =
Practice: Divide; then simplify the quotient.
1) 2) 3)
4) 5) 6)
7)
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Name_______________________________ Date__________________HW #7
Divide; then simplify the quotient.
1) 2) 3)
4) 5) 6)
7) 8) 9)
ReviewWrite 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.
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a
b
c
Legs
Hypotenuse
x
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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.
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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.
a2 + b2 = c2
3
4x
7
8
x
x
25ft15ft
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.
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x
313
x
14cm10cm
x
15in20in
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.
25
15m 8m
x
5 x
13
x
12in6in
x5
2
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!!!]
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10x
14
x9
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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?
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6cm
8cmx
20ft
x10ft
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!]
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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.
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10ft
6ft
50cm30cm
