Simplifying, Multiplying,

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Simplifying, Multiplying, Rationalizing Radicals
Homework Radical Worksheet
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Perfect Squares
64
225
1
81
256
4
100
289
9
121
16
324
144
25
400

169
36
196
49
625
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Simplify
2
4
5
This is a piece of cake!
10
12
4
Perfect Square Factor Other Factor
Simplify




LEAVE IN RADICAL FORM






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Perfect Square Factor Other Factor
Simplify




LEAVE IN RADICAL FORM






6
Simplify
7
Simplify
8
Simplify
4
9
Simplify
OR
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Combining Radicals - Addition

To combine radicals ADD the coefficients of like
radicals
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Simplify each expression
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Simplify each expression Simplify each radical
first and then combine.
Not like terms, they cant be combined
Now you have like terms to combine
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Simplify each expression Simplify each radical
first and then combine.
Not like terms, they cant be combined
Now you have like terms to combine
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Simplify each expression
 
 
 
 
 
 
 
 
 
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Simplify each expression
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Multiplying Radicals

• To multiply radicals
• multiply the coefficients
• multiply the radicands
• simplify the remaining radicals.

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Multiply and then simplify
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Squaring a Square Root
Short cut
Short cut
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Squaring a Square Root
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Dividing Radicals
To divide radicals -divide the
coefficients -divide the radicands, if possible
-rationalize the denominator so that no radical
remains in the denominator
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Rationalizing
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There is an agreement
in mathematics
that we dont leave a radical
in the denominator
of a fraction.
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So how do we change the radical denominator of a
fraction?
(Without changing the value of the fraction)
The same way we change the denominator of any
fraction
Multiply by a form of 1.
For Example
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By what number can we multiply
to change to a rational number?
The answer is . . . . . . by itself!
Squaring a Square Root gives the Root!
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Because we are changing the denominator
to a rational number,
we call this process rationalizing.
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Rationalize the denominator
(Dont forget to simplify)
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Rationalize the denominator
(Dont forget to simplify)
(Dont forget to simplify)
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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

 
 
 
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Simplify.
Divide the radicals.
Simplify.
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Simplify.

• Divide the radicals.

Uh oh There is a radical in the denominator!
Whew! It simplified!
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Simplify
Uh oh Another radical in the denominator!
Whew! It simplified again! I hope they all are
like this!
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Simplify
Uh oh There is a fraction in the radical!
Since the fraction doesnt reduce, split the
radical up.

How do I get rid of the radical in the
denominator?
Multiply by the fancy 1 to make the denominator
a perfect square!
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Fractional form of 1
This cannot be divided which leaves the radical
in the denominator. We do not leave radicals in
the denominator. So we need to rationalize by
multiplying the fraction by something so we can
eliminate the radical in the denominator.
42 cannot be simplified, so we are finished.
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Simplify fraction
Rationalize Denominator
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Use any fractional form of 1 that will result
in a perfect square
Reduce the fraction.
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Finding square roots of decimals
If a number can be made be dividing two square
numbers then we can find its square root.
For example,
3 10
12 10
0.3
1.2
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Approximate square roots
If a number cannot be written as a product or
quotient of two square numbers then its square
root cannot be found exactly.
The calculator shows this as 1.414213562
This is an approximation to 9 decimal places.
The number of digits after the decimal point is
infinite.
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Estimating square roots
10 lies between 9 and 16.
10 is closer to 9 than to 16, so ?10 will be
about 3.2
Therefore,
?9 lt ?10 lt ?16
So,
3 lt ?10 lt 4
?10 3.16 (to 2 decimal places.)
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Trial and improvement
40 is closer to 36 than to 49, so ?40 will be
about 6.3
?36 lt ?40 lt ?49
So,
6 lt ?40 lt 7
6.32
39.69
too small!
6.42
40.96
too big!
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Trial and improvement
6.332
40.0689
too big!
6.322
39.9424
too small!
Suppose we want the answer to 2 decimal places.
6.3252
40.005625
too big!
Therefore,
6.32 lt ?40 lt 6.325
?40 6.32 (to 2 decimal places)