Transcript Simplify Radicals

11-1 Simplifying
Radicals
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
6. Use a calculator to find each
square root. Round the decimal
answer to the nearest hundredth.
 46.5
6.82, -6.82
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, ...
Properties of Radicals
Multiplication property of square roots
Division property of square roots
Properties of Radicals
What does this really mean?
which we all know equals 10
it can be rewritten as:
or
which we all know equals 10
How can I use this?
To write a radical in simplest form
you must make sure:
•
•
The radicand has no perfect square factors
• The radicand has no fractions
The denominator of a fraction has no radical
This property addresses the first point
To simplify
Find a perfect square that goes into 75.
2. Simplify
Find a perfect square that goes into 600.
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
1
x x
4
x x
6
What is the answer to
x ?
x x
7
7
3
x
2
3
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.
5. Simplify 8x
12
2x
2x
25
Simplify
1.
2.
3.
4.
3x6
3x18
6
9x
18
9x
9x
36
6. Simplify 6  10
Multiply the radicals.
60
2 15
7. Simplify
Multiply the coefficients and radicals.
Simplify
1.
2.
3.
4.
4x
.
2
3
4
4 3x
2
x 48
4
48x
.
.
.
How do you know when a radical problem is done?
1. No radicals can be simplified.
Example:
not done because 4 is a factor
8
2. There are no fractions in the radical.
Example: 1 not done because it is a fraction
4
3. There are no radicals in the denominator.
Example: 1 not done because radical 5 is in
denominator
5
Division property of square roots
helps with points 2 and 3
8. Simplify.
Whew! It
simplified!
108
3
Divide the radicals.
108
3
36
6
Uh oh…
There is a
radical in the
denominator!
8
2
9. Simplify
2 8
Uh oh…
Another
radical in the
denominator!
Whew! It simplified
again! I hope they
all are like this!
2
10. Simplify
5
7
Uh oh…
There is a
fraction in
the radical!
Since the fraction doesn’t reduce, split the radical up.
5
7
How do I get rid
of the radical in
the denominator?
35

49
Multiply by the “fancy one”
to make the denominator a
perfect square!
35

7