find square roots. find cube roots. 7.1 objective the student will be able to:
TRANSCRIPT
Find square roots.
Find cube roots.
7.1 ObjectiveThe student will be able to:
If x2 = y then x is a square root of y.In the expression
is the radical sign and64 is the radicand.
1. Find the square root:
8
2. Find the square root:
-0.2
11, -11
4. Find the square root:
21
5. Find the square root:
-5/9
3. Find the square root:
The index of a cube root is always 3.
The cube root of 64 is written as .3 64
Cube Roots
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
n xindex
radicand
radical sign
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.
Perfect Cubes
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:
34 4 4 4 64
4643
because
644444 3
because
3 64 4
Examples:
3327
3
5
4
125
64
36216
3273
5
4
125
643
62163
Examples:
33 28 aa
3515 464 yy
3412 327 mm
aa 283 3
53 15 464 yy
43 12 327 mm
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.
Simplify Cube Roots
2 Extract the cube root of the factor that is a perfect cube.
1 Write the radicand as a product of two factors, where one of the factors is a perfect cube.
To simplify a cube root ...
3 The factors that are not perfect cubes will remain as the radicand.
perfect cube
Examples:
3 227 3 23
3 36 3 2125 4 a b ab 3 22 45 abba
3 1043 64 10
3 36 3 23 125 4 a a b b
3 541)
3 6402)
3 7 5500 a b3)
Not all cube roots
can be simplified!
• 30 is not a perfect cube.
• 30 does not have a perfect cube factor.
Example: 3 30
cannot be simplified! 3 30
7.2 ObjectiveThe student will be able to:
use the Pythagorean Theorem
What is a right triangle?
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.
leg
leg
hypotenuse
right angle
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 if1. a = 12 and b = 16.
122 + 162 = c2
144 + 256 = c2
400 = c2
Take the square root of both sides.
20 = c
2400 c
52 + 72 = c2
25 + 49 = c2
74 = c2
Take the square root of both sides.
8.60 = c
Find the length of the hypotenuse if2. a = 5 and b = 7.
274 c
3. Find the length of the hypotenuse given a = 6 and b = 12
1. 180
2. 324
3. 13.42
4. 18
Find the length of the leg, to the nearest hundredth, if
4. a = 4 and c = 10.42 + b2 = 102
16 + b2 = 100Solve for b.
16 - 16 + b2 = 100 - 16b2 = 84
b = 9.17
2 84b
Find the length of the leg, to the nearest hundredth, if5. c = 10 and b = 7.
a2 + 72 = 102
a2 + 49 = 100Solve for a.a2 = 100 - 49
a2 = 51
a = 7.14
2 51a
6. Find the length of the missing side given a = 4 and c = 5
1. 1
2. 3
3. 6.4
4. 9
7. The measures of three sides of a triangle are given below. Determine whether each
triangle is a right triangle. , 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.
9 + 64 = 7373 = 73
Since this is true, the triangle is a right triangle!! If it was not true, it
would not be a right triangle.
Sides: , 3, and 832 + 82 = ( ) 2
8. Determine whether the triangle is a right triangle given the sides 6, 9, and 45
1. Yes
2. No
3. Purple
7.3 ObjectivesThe student will be able to:
1. simplify square roots, and
2.simplify radical expressions.
1 • 1 = 12 • 2 = 43 • 3 = 9
4 • 4 = 165 • 5 = 256 • 6 = 36
49, 64, 81, 100, 121, 144, ...
What numbers are perfect squares?
1. Simplify
Find a perfect square that goes into 147.
147 7 3
2. Simplify
Find a perfect square that goes into 605.
11 5
Simplify
1. .
2. .
3. .
4. .
2 18
72
3 8
6 236 2
Look at these examples and try to find the pattern…
How do you simplify variables in the radical?
1x x2x x3x x x4 2x x5 2x x x6 3x x
What is the answer to ?
7 3x x x
As a general rule, divide the exponent by two. The remainder stays in the
radical.
Find a perfect square that goes into 49.
4. Simplify
7x
5. Simplify 258x
122 2x x
Simplify 369x
1. 3x6
2. 3x18
3. 9x6
4. 9x18
Multiply the radicals.
7. Simplify
60
4 154 152 15
8. Simplify Multiply the coefficients and radicals.
6 294
6 49 66 649
42 6
6 67
9.Simplify
1. .
2. .
3. .
4. .
24 3x44 3x
2 48x448x
36 8x x
How do you know when a radical problem is done?
1. No radicals can be simplified.Example:
2. There are no fractions in the radical.Example:
3. There are no radicals in the denominator.Example:
8
1
4
1
5
10. Simplify.
Divide the radicals.
108
3
366
Uh oh…There is a
radical in the denominator!
Whew! It simplified!
11. Simplify
4 1
4
4
2
2
Uh oh…Another
radical in the denominator!
Whew! It simplified again! I hope they all are like this!
12. Simplify
5
7
5
7
75
7 7
35
49 35
7
Since the fraction doesn’t reduce, split the radical up.
Uh oh…There is a fraction in the radical!
How do I get rid of the radical in
the denominator?
Multiply by the “fancy one” to make the denominator a
perfect square!
7.4 ObjectiveThe student will be able to:
simplify radical expressions involving addition and subtraction.
1. Simplify.
Just like when adding variables, you can only combine LIKE radicals.
5 5
Which are like radicals?
2. Simplify.
4 7 3 3
Simplify
1. .
2. .
3. .
4. .
5 2 6 2 4 2
5 2 6 2 4 2 15 2
3 2
7 2
Perimeter = Add all of the sides
3. Find the perimeter of a rectangle whose length is and whose
width is
8 6 6 3 8
Simplify each radical.
4. Simplify.
9 164 3 2 3 42 5
3 44 3 2 3 22 5
12 3 8 3 4 5
4 3 4 5
Combine like radicals.
5. Simplify
25 368 2 5 2 2 249 5 68 2 5 2 72 2
40 2 30 2 14 2 56 2
Simplify
1. .
2. .
3. .
4. .
5 3 4 2 3 3
5 3 4 2 3 3 6 22 3 4 28 3 4 2
Simplify
1. .
2. .
3. .
4. .
3 12 4 27
7 39
48 3
48 6
18 3