square roots. this isn’t exactly true, but for the next 3 weeks: “radical” means the same...
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SQUARE ROOTS
This isn’t exactly true, but for the next 3 weeks:
“Radical” means the same thing as “square
root”
*Side Note:
What is a square root?
What is a square root?
A square root is a way of asking:“What number, times itself gives this value?”
What is a square root?
A square root is a way of asking:“What number, times itself gives this value?”
25 “What number times itself is 25?”
What is a square root?
A square root is a way of asking:“What number, times itself gives this value?”
25 “What number times itself is 25?”
525 “5 times 5 is 25.”
25 “What number times itself is 25?”
525 “5 times 5 is 25.”
* This is called “taking the square root.”* Notice, after we have taken the square root, the symbol is gone.
14436
12125
10016
819
644
491
These are the numbers that have a “nice” answer when we take a square root:
14436
12125
10016
819
644
491 1 What number times itself is 1?
These are the numbers that have a “nice” answer when we take a square root:
14436
12125
10016
819
644
491 1
2 What number times itself is 4?
These are the numbers that have a “nice” answer when we take a square root:
14436
12125
10016
819
644
491 1
2
3 What number times itself is 9?
These are the numbers that have a “nice” answer when we take a square root:
14436
12125
10016
819
644
491 1
2
3
4 What number times itself is 16?
These are the numbers that have a “nice” answer when we take a square root:
14436
12125
10016
819
644
491 1
2
3
4
5 What number times itself is 25?
These are the numbers that have a “nice” answer when we take a square root:
These are the numbers that have a “nice” answer when we take a square root:
14436
12125
10016
819
644
491 1
2
3
4
5
6 What number times itself is 36?
These are the numbers that have a “nice” answer when we take a square root:
These are the numbers that have a “nice” answer when we take a square root:
14436
12125
10016
819
644
491 1
2
3
4
5
6
7
8
9
10
11
12
Since these numbers give nice answers, we call them perfect squares.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
Since these numbers give nice answers, we call them perfect squares.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
Since these numbers give nice answers, we call them perfect squares.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
You will have to know these numbers.
Since these numbers give nice answers, we call them perfect squares.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
You will have to know these numbers.Do I have to memorize
them?
Since these numbers give nice answers, we call them perfect squares.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
You will have to know these numbers.Do I have to memorize
them?
You can, but…It is probably easier to just remember how we
get them:
Since these numbers give nice answers, we call them perfect squares.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
You will have to know these numbers.Do I have to memorize
them?
You can, but…It is probably easier to just remember how we
get them:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
12 22 32 42 52 62 72 82 92 102 112
122
Just square the numbers 1-12.
So, if you have a problem that looks like this:
?16
So, if you have a problem that looks like this:
?16
Just write the number that answers the question:
So, if you have a problem that looks like this:
?16
Just write the number that answers the question:“What number times itself is 16?”
So, if you have a problem that looks like this:
?16
Just write the number that answers the question:“What number times itself is 16?”
4
The hard part is when the answer is not obvious:
20
The hard part is when the answer is not obvious:
20
No nice number times itself is 20.
The hard part is when the answer is not obvious:
20
No nice number times itself is 20.
There are two options for how to find the square root of 20.
The hard part is when the answer is not obvious:
20
No nice number times itself is 20.
Option 1 Get an approximate answer (decimal) You just type it in the calculator.
(easy)
There are two options for how to find the square root of 20.
The hard part is when the answer is not obvious:
20
No nice number times itself is 20.
Option 1 Get an approximate answer (decimal) You just type it in the calculator.
(easy)
Option 2 The exact answer. This is like reducing a fraction (hard)
There are two options for how to find the square root of 20.
The hard part is when the answer is not obvious:
20
No nice number times itself is 20.
Option 1 Get an approximate answer (decimal) You just type it in the calculator.
(easy)
Option 2 The exact answer. This is like reducing a fraction (hard)
There are two options for how to find the square root of 20.
You will have to be able to do BOTH.
Option 1 Get an approximate answer (decimal) You just type it in the calculator.
(easy)
20
Option 1 Get an approximate answer (decimal) You just type it in the calculator.
(easy)
20
Use the square root button here to just type it
in the calculator.
20
Option 1 Get an approximate answer (decimal) You just type it in the calculator.
(easy)
20
Use the square root button here to just type it
in the calculator.
Did you get 400?
Make sure to hit the “ctrl” key first!
20
Option 1 Get an approximate answer (decimal) You just type it in the calculator.
(easy)
20
Use the square root button here to just type it
in the calculator.
2047213595.4
Did you get 400?
Make sure to hit the “ctrl” key first!
Option 1 Get an approximate answer (decimal) You just type it in the calculator.
(easy)
2020
47213595.4
This isn’t the best answer, because the
calculator had to round the answer.
Option 1 Get an approximate answer (decimal) You just type it in the calculator.
(easy)
2020
47213595.4
This isn’t the best answer, because the
calculator had to round the answer.
That’s why we call it “approximate”
Option 2 Get an EXACT answer
20
A square root is a lot like a fraction.
There are many different ways to write the same number.
Option 2 Get an EXACT answer
20
A square root is a lot like a fraction.
There are many different ways to write the same number.
For example:
Option 2 Get an EXACT answer
20
A square root is a lot like a fraction.
There are many different ways to write the same number.
For example:
2
1
4
2
6
3
000,12
000,6
150
75
100
50
20
10
Option 2 Get an EXACT answer
20
For example:
2
1
4
2
6
3
000,12
000,6
150
75
100
50
20
10
2105452
Option 2 Get an EXACT answer
20
For example:
2
1
4
2
6
3
000,12
000,6
150
75
100
50
20
10
2105452
Just like with fractions, all these are the same thing. But they are not reduced.
Option 2 Get an EXACT answer
20
For example:
2
1
4
2
6
3
000,12
000,6
150
75
100
50
20
10
2105452
Like fractions, there is a BEST ANSWERWe call it a reduced radical
Option 2 Get an EXACT answer
20
Option 2 Get an EXACT answer
20
You learned how to reduce the radical in algebra 1
Option 2 Get an EXACT answer
20
You learned how to reduce the radical in algebra 1There are two good ways:
Option 2 Get an EXACT answer
20
You learned how to reduce the radical in algebra 1There are two good ways:
Method 1 the factor tree (this is easy, but takes a long time)https://www.youtube.com/watch?v=jC5-2FOCXGg
Method 2 using perfect squares. (Faster, but not as easy to remember) https://www.youtube.com/watch?v=sOmnOQn5HsU
Option 2 Get an EXACT answer
20
You learned how to reduce the radical in algebra 1There are two good ways:
Method 1 the factor tree (this is easy, but takes a long time)https://www.youtube.com/watch?v=jC5-2FOCXGg
Method 2 using perfect squares. (Faster, but not as easy to remember) https://www.youtube.com/watch?v=sOmnOQn5HsU
Watch one or both of the videos
Option 2 Get an EXACT answer
20
You learned how to reduce the radical in algebra 1There are two good ways:
Method 1 the factor tree (this is easy, but takes a long time)https://www.youtube.com/watch?v=jC5-2FOCXGg
Method 2 using perfect squares. (Faster, but not as easy to remember) https://www.youtube.com/watch?v=sOmnOQn5HsU
I don’t get it! And I hate that guy in
the video. What do I do?
Option 2 Get an EXACT answer
20
You learned how to reduce the radical in algebra 1There are two good ways:
Method 1 the factor tree (this is easy, but takes a long time)https://www.youtube.com/watch?v=jC5-2FOCXGg
Method 2 using perfect squares. (Faster, but not as easy to remember) https://www.youtube.com/watch?v=sOmnOQn5HsU
I don’t get it! And I hate that guy in
the video. What do I do?
A) Ask Mr. B for help .B) Type “simplify square
roots” into youtube.15,000 hits!
#1 on your worksheet
20
Here is how to do it
#1 on your worksheet
20 20
Here is how to do it
#1 on your worksheet
20 20
102
Here is how to do it
#1 on your worksheet
20 20
102
5 2
Here is how to do it
#1 on your worksheet
20 20
102
5 2
522
Here is how to do it
#1 on your worksheet
20 20
102
5 2
522
Here is how to do it
#1 on your worksheet
20 20
102
5 2
522
52
Here is how to do it
Or do it this way:
#1 on your worksheet
20
#1 on your worksheet
20
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144Perfect squares:
#1 on your worksheet
20
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144Perfect squares:
Find the biggest perfect square you can divide 20
by and NOT get a decimal
#1 on your worksheet
20
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144Perfect squares:
Find the biggest perfect square you can divide 20
by and NOT get a decimalIt’s 4
#1 on your worksheet
20
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144Perfect squares:
Find the biggest perfect square you can divide 20
by and NOT get a decimalIt’s 4
Rewrite the square root using the perfect square
times a number
#1 on your worksheet
20
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144Perfect squares:
Find the biggest perfect square you can divide 20
by and NOT get a decimalIt’s 4
Rewrite the square root using the perfect square
times a number
54
#1 on your worksheet
20
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144Perfect squares:
Find the biggest perfect square you can divide 20
by and NOT get a decimalIt’s 4
Rewrite the square root using the perfect square
times a number
Take the square root of the perfect square.
54
#1 on your worksheet
20
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144Perfect squares:
Find the biggest perfect square you can divide 20
by and NOT get a decimalIt’s 4
Rewrite the square root using the perfect square
times a number
Take the square root of the perfect square.
54
52
#1 on your worksheet
20
DONE
54
52
You should now be able to do problems 1-12U
se e
ither
meth
od
20
Other side of your paper(multiplying square roots)
Part 2 Multiplying square roots
Part 2 Multiplying square roots
When we multiply two square roots
Part 2 Multiplying square roots
When we multiply two square roots
Multiply the numbers in the square root102
Part 2 Multiplying square roots
When we multiply two square roots
Multiply the numbers in the square root102
20
Part 2 Multiplying square roots
102
20
Part 2 Multiplying square roots
102
20Then, we have to see if this square root can be
reduced
Part 2 Multiplying square roots
102
20(this is just like #1)
Part 2 Multiplying square roots
102
20(this is just like #1)
52
Now, do problems 13-15
For this type of problem, we have regular numbers AND
square roots being multiplied
For this type of problem, we have regular numbers AND
square roots being multiplied
Multiply the regular numbers first
For this type of problem, we have regular numbers AND
square roots being multiplied
Multiply the regular numbers first15
For this type of problem, we have regular numbers AND
square roots being multiplied
Multiply the regular numbers first15
For this type of problem, we have regular numbers AND
square roots being multiplied
Multiply the regular numbers first15
Next, multiply the square roots
For this type of problem, we have regular numbers AND
square roots being multiplied
Multiply the regular numbers first15
Next, multiply the square roots
For this type of problem, we have regular numbers AND
square roots being multiplied
Multiply the regular numbers first15
Next, multiply the square roots
12
For this type of problem, we have regular numbers AND
square roots being multiplied
Multiply the regular numbers first15
Next, multiply the square roots
12
Then, simplify the square root.
For this type of problem, we have regular numbers AND
square roots being multiplied
Multiply the regular numbers first15
Next, multiply the square roots
12
Then, simplify the square root.
32
For this type of problem, we have regular numbers AND
square roots being multiplied
Multiply the regular numbers first15
Next, multiply the square roots
12
Then, simplify the square root.
32
For this type of problem, we have regular numbers AND
square roots being multiplied
Multiply the regular numbers first15
Next, multiply the square roots
12
Then, simplify the square root.
32
Bring back the regular number
For this type of problem, we have regular numbers AND
square roots being multiplied
Multiply the regular numbers first15
Next, multiply the square roots
12
Then, simplify the square root.
32
Bring back the regular number
15
For this type of problem, we have regular numbers AND
square roots being multiplied
Multiply the regular numbers first15
Next, multiply the square roots
12
Then, simplify the square root.
32
Bring back the regular number
15
Finally, re-multiply the regular numbers
For this type of problem, we have regular numbers AND
square roots being multiplied
Multiply the regular numbers first15
Next, multiply the square roots
12
Then, simplify the square root.
32
Bring back the regular number
15
Finally, re-multiply the regular numbers
330
Now, do problems 16-18
This is an easy one.
This is an easy one.
“squared” means multiply be itself.
This is an easy one.
“squared” means multiply be itself.
88
This is an easy one.
“squared” means multiply be itself.
88
64
This is an easy one.
“squared” means multiply be itself.
88
What number times itself is 64?64
This is an easy one.
“squared” means multiply be itself.
88
What number times itself is 64?64
8
This is an easy one.
“squared” means multiply be itself.
88
What number times itself is 64?64
8 There is a very simple pattern to these problems
Do problems 19-21(Yes, they are very easy)
This is just like the last one, but there is a regular number
too!
2525
This is just like the last one, but there is a regular number
too!
Write it out
2525
This is just like the last one, but there is a regular number
too!
2255
Write it out
2525
This is just like the last one, but there is a regular number
too!
2255
Write it out
Multiply the regular numbers
2525
This is just like the last one, but there is a regular number
too!
2255
25
Write it out
Multiply the regular numbers
2525
This is just like the last one, but there is a regular number
too!
2255
25
Write it out
Multiply the regular numbers
Multiply the square roots
2525
This is just like the last one, but there is a regular number
too!
2255
25 2
Write it out
Multiply the regular numbers
Multiply the square roots
2525
This is just like the last one, but there is a regular number
too!
2255
25 2
50
Write it out
Multiply the regular numbers
Multiply the square roots
Do problems 22-24(there is an easy pattern here, too.)
Rationalizing the denominator
2
6 This is BAD.
We are not allowed to have a square root in the denominator
(bottom of a fraction)
We will have problems that end like this, and we have to know how to fix ‘em
How to get rid of a square root on the bottom of a fraction
You can’t leave “root 2” in the bottom of the fraction
What is “root 2” times “root 2”?
This is how you have to leave your answers
2
3
2
2
22
23
2
23
YOU DON’T NEED TO WRITE THIS, THIS IS JUST AN EXAMPLE
5
7
5
5
11
2
3
5
5
57
2
2 22
2
3
3
3
35
2
82
2
2
28 24
5
105
5
5
510 52
1.
2.
3.
4.
5.
YOU DON’T NEED TO WRITE THESE, THEY ARE JUST EXAMPLES
Menu
To get rid of the square root on the bottom of a fraction…
To get rid of the square root on the bottom of a fraction…
…Multiply the top and bottom of the fraction by the square root you are trying to get rid
of.
To get rid of the square root on the bottom of a fraction…
…Multiply the top and bottom of the fraction by the square root you are trying to get rid
of.
2
2
To get rid of the square root on the bottom of a fraction…
…Multiply the top and bottom of the fraction by the square root you are trying to get rid
of.
2
2
2
26
To get rid of the square root on the bottom of a fraction…
…Multiply the top and bottom of the fraction by the square root you are trying to get rid
of.
2
2
2
26
Then reduce the regular numbers
To get rid of the square root on the bottom of a fraction…
…Multiply the top and bottom of the fraction by the square root you are trying to get rid
of.
2
2
2
26
Then reduce the regular numbers
To get rid of the square root on the bottom of a fraction…
…Multiply the top and bottom of the fraction by the square root you are trying to get rid
of.
2
2
2
26
Then reduce the regular numbers
23
Do problems 25-28
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