Radical Expressions

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Radical Expressions

• MA.912.A.6.1 Simplify radical expressions.
• MA.912.A.6.2 Add, subtract, multiply, and divide
  radical expressions

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What is a radical?
Radical
Vinculum
This is the symbol for square root. The
implied index is 2, however it does not need to
be written.
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Root Index
If the index is 3, it is called a 3rd root, or
cube root.
Radicand
3
nth roots
Index Symbol Name
2 Square Root
3 Cube Root
4 4th Root
n nth Root
Notice how the index is invisible in the square
root.
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What is a square root?
The number c is a square root of the number a
if c2 a.
The definition is saying that the number 6 is a
square root of 36, since 62 36. But since (-
6)2 (- 6)(- 6) 36 , it follows by the above
definition that - 6 is also a square root of
36. Thus, 36 has two square roots a positive
one and a negative one.
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What is a square root?
The notation used to indicate the square root of
some number a is . When a square root
is written with the radical symbol we are
referring to the positive, or principal square
root. For example, 6 . We will
adopt the notation of to represent the
negative square root of a number a. For
example, - 6 . When using
words What is the square root of, then the
answer should indicate both roots.
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Perfect Squares
121, 22 4, 32 9, 42 16, 52 25, 62 36, 72
49, 82 64, 92 81, 102 100, 112 121, 122
144, 132 169, 142 196, 152 225 202
400
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Evaluate
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Area 64 sq units
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Approximating a Square Root
Approximate the square root to the nearest integer
122 144 132 169
Since 149 is between 144 and 169, the square root
of 149 is Between 12 and 13.
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Approximating a Square Root
Approximate the square root to the nearest integer
72 49 82 64
Since 57 is between 49 and 64, the square root of
57 is Between 7 and 8.
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Estimate the square root of 33

• You know that it is going to be between 5 and 6
  since 33 falls between 25 and 36.
• Draw a 5 x 5 box to represent the square root of
  25. Shade in the 25 boxes.

• Add on another row and column so that you now
  have
• 6 x 6 box to represent the
• square root of 36.

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Estimate the square root of 33

• Shade in the extra boxes so you have 33 colored
  in.

• 8 out of the 11 extra boxes are shaded, this
  represents 8/11

• The square root of 33 is
• estimated to

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Estimate the square root of 85

• You know that it is going to be between 9 and 10
  since 85 falls between 81 and 100.
• Draw a 9 x 9 box to represent the square root of
  81. Shade in the 81 boxes.
• Add on another row and column so that you now
  have
• 10 x 10 box to represent the
• square root of 100.

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Estimate the square root of 85

• Shade in the extra boxes so you have 85 colored
  in.

• 4 out of the 19 extra boxes are
• shaded, this represents 4/19
• The square root of 85 is
• estimated to

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Simplify
Click link below to graph
or see next slide.
http//my.hrw.com/math06_07/nsmedia/tools/Graph_Ca
lculator/graphCalc.html
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x F(x)
0 0
1 1
-1 1
2 2
-2 2
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(No Transcript)
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Simplify
NO! When evaluating an even root and the
resulting answer has an even exponent, absolute
value symbols are not necessary.
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Product Property of Radicals
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Simplify
The following statement is found in most
textbooks In this book, all variables in
radical expressions represent non-negative
numbers This statement eliminates the need
to use the absolute value symbol.
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Simplifying Square Roots
We can use the Product Property of Radicals to
simplify radical expressions.
Look for a perfect square factor
A square root is simplified if there are no
perfect square factors in the radicand.
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Simplifying Square Roots
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Simplifying Square Roots
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Simplifying Square Roots
You can use divisibility rules and prime
factoring to help with simplifying square roots.
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363
3
121
11
11
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Multiplying Radicals
or
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Multiplying Radicals
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Multiplying Radicals
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Multiplying Radicals
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Quotient Property of Radicals
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Dividing Radicals
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Dividing Radicals
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Simple Radical FormA radical is said to be in
simple radical form if

• There are no radicals in the denominator
• The radicand(s) in the numerator are as small as
  possible.

Not simplified radical form
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The Process of eliminating a radical from the
denominator is called
Rationalizing the Denominator

• To rationalize the denominator, you must convert
  the radicand into a perfect square number. In
  this case you can do it by multiplying by ____ .
• In order not to change the value of the original
  number, you must multiply the numerator by_____
  as well.

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Rationalizing the Denominator
Why?
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Express in simple radical form
reduce
Now Rationalize!
reduce
Sometimes it is more efficient to simplify the
radical before you rationalize.
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Express in simple radical form
Sometimes it is more efficient to reduce the
fraction before you rationalize.
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Like Radicals
Two radicals are called like radicals if they
have the same index and radicand.
Are the examples below like radicals?
NO, the indexes are not the same.
YES, the radicals have the same index and
radicand.
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Adding Subtracting Radicals

• When adding or subtracting radicals, you will
  use the same concept as that of adding or
  subtracting "like" variables.
•  
• In other words, the radicals must be like
  radicals before you add (or subtract) them.

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Adding Subtracting Radicals
Procedure
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Adding Subtracting Radicals
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(No Transcript)
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Simplify the radical expression
or
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Simplify the radical expression
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Simplify the radical expression
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Simplify the radical expression
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EOC Practice
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EOC Practice
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Game Time!
http//www.math-play.com/square-roots-game.html