SIMPLIFYING SQUARE ROOTS:
Remember: When we simplify a square root we look
for the largest perfect square which is a factor of the given number.
LECTURE ON SIMPLYFING SQUARE ROOTS
| Example:
Simplify 
Even though 4 is a perfect square that is a factor or 48, the largest
perfect square
which is a factor if 48 is 16. If we used 4 we would have
to simplify twice instead of
only once.
Using 4 we would have:
THIS IS MUCH TOO MUCH WORK!!!
Here is the better way:
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Simplify the following square roots:
1.  |
2.  |
3.  |
4.  |
5.  |
6.  |
7.  |
8.  |
9.  |
Answers:
1.  |
2.  |
3.  |
4.  |
5.  |
6.  |
7.  |
8. not a real number |
9.  |
MULTIPLYING and SIMPLIFYING SQUARE ROOTS:
LECTURE ON MULTIPLYING SQUARE ROOTS
Example: Multiply
and Simplify: 
This can be done in two different ways:
THE HARDER WAY!!
Now you are left with trying to find
perfect squares that are factors of this
number. This can be difficult.
We know 9 goes into 4050 by the
divisibility rule for 9 (9 goes into a
number if the sum of the digits of the
number is divisible by 9).
The sum of 4 + 0 + 5 + 0 is 9.
Since 9 is divisible by 9, 9 goes into 4050.
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THE EASIER WAY!!
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Multiply and Simplify:
1.  |
2.  |
3.  |
4.  |
5.  |
6.  |
7.  |
8.  |
9.  |
Answers:
1.  |
2.  |
3.  |
4.  |
5.  |
6.  |
7.  |
8.  |
9.  |
ADDING and SUBTRACTING SQUARE ROOTS
Just as we must have like terms when adding or subtracting
algebraic expressions
we must have like square roots when we add or subtract square
roots.
LECTURE ON ADDING SQUARE ROOTS
Example: Find the answer to 
The square roots in this problem are not like square roots.
First we need to simplify each square root.
 |
Find the answers to each of the following:
1.  |
2.  |
3.  |
4.  |
5.  |
6.  |
7.  |
8.  |
9.  |
10.  |
11.  |
12.  |
13.  |
14.  |
15.  |
16.  |
17.  |
18.  |
Answers:
1.  |
2.  |
3. |
4.  |
5.  |
6.  |
7.  |
8.  |
9.  |
10.  |
11.  |
12.  |
| 13. 0 |
14. 0 |
15.  |
16.  |
17.  |
18.  |
SOLVING QUADRATIC EQUATIONS USING THE
SQUARE ROOT PROPERTY
LECTURE ON THE SQUARE ROOT PROPERTY
Example 1:
The way we are used to solving this equation is to set the equation
equal to zero.
x 2 - 25 =
0
Next we factor:
(x - 5) (x + 5) = 0
x - 5 = 0
x = 5 |
x + 5 = 0
x = - 5 |
The Solution Set is {- 5, 5}.
Another way we can write this is { }
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When we have an equation involving the difference of squares (i.e. x
2 - a 2 =
0)
we should realize that we will always get an answer that is always
plus and minus
the square root of a
2 .
In the case of Example 1 the answer is: - or + or
- 5 and + 5
This fact can be used to solve the equation
x 2 = 25
by using the SQUARE ROOT PROPERTY.
The SQUARE ROOT PROPERTY states
that if we have an equation of the form:
x 2 = a
then x = 
or x = -
|
In solving x
2 = 25 we have:
x = -?
or +
which we can write x = 
In Solution Set form: { }
| Example 2:
We can now use the SQUARE ROOT PROPERTY to
solve equations that we could not solve before.
Solve: x 2 = 19
If we set this equal to zero we can see that this is not a
difference of squares, however, by the SQUARE
ROOT PROPERTY:
x = 
or x = -
or
x = 
In Solution Set form { } |
**************************************************
| Example 3:
Solve: x 2 = 20
We now know that x =
or x = - or
Simplifying: x = 
or
x = 
so x = 
In Solution Set form { }
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Solve each of the following equations.
Express radicals in simplest form.
1.  |
2.  |
3.  |
4.  |
5.  |
6.  |
7.  |
8.  |
9.  |
10.  |
11.  |
12.  |
Answers:
1. { 8} |
2. {10} |
3. { } |
4. { } |
5. { } |
6. { } |
7. { } |
8. { } |
9. { } |
10. { } |
11. { } |
12. { } |
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© Howard Sorkin
2016 All rights reserved |
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