Transcript Notes 13 - Henry Ford Algebra

7.1 Objective
The student will be able to:
Find square roots.
Find cube roots.
If x2 = y then x is a square root of y.
In the expression 64
is the radical sign and
64 is the radicand.
1. Find the square root: 64
8
2. Find the square root: - 0.04
-0.2
3. Find the square root: ± 121
11, -11
4. Find the square root:
21
5. Find the square root:
-5/9
441
25
81
Cube Roots
The index of a cube root is
always 3.
The cube root of 64 is written as
3
64 .
What does cube root mean?
The cube root of a number is…
…the value when multiplied by itself
three times gives the original number.
Cube Root Vocabulary
radical sign
index
n
x
radicand
Perfect Cubes
If a number is a perfect cube, then
you can find its exact cube root.
A perfect cube is a number that can
be written as the cube (raised to
third power) of another number.
What are Perfect Cubes?
• 13 = 1 x 1 x 1 = 1
• 23 = 2 x 2 x 2 = 8
• 33 = 3 x 3 x 3 = 27
• 43 = 4 x 4 x 4 = 64
• 53 = 5 x 5 x 5 = 125
• and so on and on and on…..
Examples:
3
64  4
because
 4 4 4    4 
3
64   4
 4 4 4
3
 64
because

 4
3
  64
Examples:
27  3
3
216  6
3
64  4 
 
125  5 
3
3
3
3
27  3
216  6
64
4

125 5
Examples:
8a  2a 

27m  3m
12
 64 y
15
3
3
3


3

3
4 3
  4y
5 3
8a  2a
3
27m
 3m
 64 y
  4y
12
15
4
5
Simplify Cube Roots
Not all numbers or expressions have an
exact cube root as in the previous
examples.
If a number is NOT a perfect cube,
then you might be able to SIMPLIFY it.
To simplify a cube root ...
1 Write the radicand as a product of
two factors, where one of the
factors is a perfect cube.
2 Extract the cube root of the
factor that is a perfect cube.
3 The factors that are not perfect
cubes will remain as the radicand.
Examples:
perfect cube
1)
3
54 
2)
3
640 
3)
3
3
3
27  2  33 2
3
64  10  4 10
3
500a b  125  4  a a  b b 
7
5
3
3
6
3
3
125a b  4ab  5a b 4ab
6
3
3
2
2 3
2
2
Not all cube roots
can be simplified!
Example:
3
30
• 30 is not a perfect cube.
• 30 does not have a perfect cube
factor.
3
30 cannot be simplified!
7.2 Objective
The student will be able to:
use the Pythagorean Theorem
What is a right triangle?
hypotenuse
leg
right angle
leg
It is a triangle which has an angle that is 90
degrees.
The two sides that make up the right angle
are called legs.
The side opposite the right angle is the
hypotenuse.
The Pythagorean Theorem
In a right triangle, if a and b are the
measures of the legs and c is the
hypotenuse, then
a2 + b2 = c2.
Note: The hypotenuse, c, is always
the longest side.
Find the length of the
hypotenuse if
1. a = 12 and 2b = 16.
2
2
12 + 16 = c
144 + 256 = c2
400 = c2
Take the square root of both sides.
2
400  c
20 = c
Find the length of the hypotenuse if
2. a = 5 and b = 7.
5 2 + 7 2 = c2
25 + 49 = c2
74 = c2
Take the square root of both sides.
74  c
2
8.60 = c
3. Find the length of the hypotenuse
given a = 6 and b = 12
1.
2.
3.
4.
180
324
13.42
18
Find the length of the leg, to the
nearest hundredth, if
4. a = 4 and c = 10.
42 + b2 = 102
16 + b2 = 100
Solve for b.
16 - 16 + b2 = 100 - 16
b2 = 84
2
b  84
b = 9.17
Find the length of the leg, to the
nearest hundredth, if
5. c = 10 and b = 7.
a2 + 72 = 102
a2 + 49 = 100
Solve for a.
a2 = 100 - 49
a2 = 51
2
a  51
a = 7.14
6. Find the length of the missing side
given a = 4 and c = 5
1.
2.
3.
4.
1
3
6.4
9
7. The measures of three sides of a triangle
are given below. Determine whether each
triangle is a right triangle.
73 , 3, and 8
Which side is the biggest?
The square root of 73 (= 8.5)! This must be
the hypotenuse (c).
Plug your information into the Pythagorean
Theorem. It doesn’t matter which number
is a or b.
Sides: 73 , 3, and 8
32 + 82 = ( 73 ) 2
9 + 64 = 73
73 = 73
Since this is true, the triangle is a
right triangle!! If it was not true, it
would not be a right triangle.
8. Determine whether the triangle is a right
triangle given the sides 6, 9, and 45
1. Yes
2. No
3. Purple
7.3 Objectives
The student will be able to:
1. simplify square roots, and
2. simplify radical expressions.
What numbers are perfect squares?
1•1=1
2•2=4
3•3=9
4 • 4 = 16
5 • 5 = 25
6 • 6 = 36
49, 64, 81, 100, 121, 144, ...
1. Simplify
147
Find a perfect square that goes into 147.
147  7 3
2. Simplify 605
Find a perfect square that goes into 605.
11 5
Simplify
1.
2.
3.
4.
2 18
.
3 8
6 2
36 2
.
.
.
72
How do you simplify variables in the radical?
x
7
Look at these examples and try to find the pattern…
x  x
2
x x
3
x x x
4
2
x x
5
2
x x x
6
3
x x
1
What is the answer to
x x
7
3
x ?
7
x
As a general rule, divide the
exponent by two. The
remainder stays in the
radical.
4. Simplify 49x
2
Find a perfect square that goes into 49.
7x
5. Simplify 8x
12
2x
2x
25
Simplify
1.
2.
3.
4.
3x6
3x18
6
9x
18
9x
9x
36
7. Simplify 6 · 10
Multiply the radicals.
60
4 15
4
15
2 15
8. Simplify 2 14 · 3 21
Multiply the coefficients and radicals.
6 294
6 49 6
6
49
67
6
6
42 6
9.Simplify 6 x
1.
2.
3.
4.
4x
.
2
3
4
4 3x
2
x 48
4
48x
.
.
.
3
8x
How do you know when a radical
problem is done?
1. No radicals can be simplified.
Example:
8
2. There are no fractions in the radical.
1
Example:
4
3. There are no radicals in the denominator.
Example:
1
5
10. Simplify.
Whew! It
simplified!
108
3
Divide the radicals.
108
3
36
6
Uh oh…
There is a
radical in the
denominator!
8
2
11. Simplify
2 8
4 1
4
Whew! It simplified
again! I hope they
all are like this!
4
2
2
Uh oh…
Another
radical in the
denominator!
12. Simplify
5
7
Uh oh…
There is a
fraction in
the radical!
Since the fraction doesn’t reduce, split the radical up.
5
7
5

7
How do I get rid
of the radical in
the denominator?
7
7
35

49
Multiply by the “fancy one”
to make the denominator a
perfect square!
35

7
7.4 Objective
The student will be able to:
simplify radical expressions
involving addition and subtraction.
1. Simplify.
3 5+4 5-2 5
Just like when adding variables, you
can only combine LIKE radicals.
5 5
2. Simplify. 6 7 - 3 - 2 7 + 4 3
Which are like radicals?
4 7 3 3
Simplify 5 2  6 2  4 2
1.
2.
3.
4.
5 2 6 2 4 2
15 2
3 2
7 2
.
.
.
.
3. Find the perimeter of a rectangle
whose length is 4 6 + 3 and whose
width is 2 3 - 4.
4 6+ 3
2 3 - 4.
2 3 - 4.
4 6+ 3
Perimeter = Add all of the sides
8 6  6 3 8
4. Simplify. 4 27 - 2 48 + 2 20
Simplify each radical.
4 9 3  2 16 3  2 4 5
4 3 32 4 32 2 5
12 3  8 3  4 5
Combine like radicals.
4 34 5
5. Simplify 8 50 + 5 72 - 2 98
8 25 2  5 36 2  2 49 2
8 5 2 5 6 2 2 7 2
40 2  30 2 14 2
56 2
Simplify 5 3  4 2  3 3
1.
2.
3.
4.
5
6
2
8
.
.
.
3  4 2 3 3
2
34 2
34 2
.
Simplify 3 12  4 27
1.
2.
3.
4.
7 39
48 3
48 6
18 3
.
.
.
.