RADICAL EXPRESSIONS

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RADICAL EXPRESSIONS
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RADICAL EXPRESSIONS
ARE THE SAME AS SQUARE ROOTS
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Radical Expressions

• Square Roots Perfect Squares
• Simplifying Radicals
• Multiplying Radicals
• Dividing Radicals
• Adding Radicals

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What is the square root?
64
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The square root of a number is a when
multiplied by itself equals the number
64
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8
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What is the square root?
X2
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A Square Root is a term when squared is the
Perfect Square
X2
X
X
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What is the square root?
64x2
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Answer
64x2
8x
8x
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What is the square root?
4x2
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Answer
4x2
2x
2x
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What is the square root?
(x3)2
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Answer
(x3)
(x3)2
(x3)
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Radical Sign
This is the symbol for square root If the
number 3 is on top of the v, it is called the 3rd
root.
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Radical Sign
This is the 4th root. This is the 6th
root If n is on top of the v, it is called
the nth root.
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Cube Root
If the sides of a cube are 3 inches Then the
volume of the cube is 3 times 3 times 3 or 33
which is 27 cubic inches.
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NUMBER Perfect Squares
Numbers that are perfect squares are 121, 22
4, 32 9, 42 16, 52 25, 62 36, 72 49, 82
64, 92 81, 102 100,
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Recognizing Perfect Squares(NAME THE SQUARE
ROOTS)
Variables that are perfect squares are x2, a4,
y22, x100 (Any even powered variable is a
perfect square)
x25
EVEN POWERED EXPONENTS ARE SQUARES
x50
x25
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NAME THE SQUARE ROOT
9x50
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3x25
EVEN POWERED EXPONENTS ARE SQUARES
9x50
3x25
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Recognizing Perfect Squares(NAME THE SQUARE
ROOTS)
x2
4
4x6
25x2
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Recognizing Perfect Squares(NAME THE SQUARE
ROOTS)
x
2
x
x2
2
4
5x
2x3
5x
2x3
4x6
25x2
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Simplifying Square Roots
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Simplifying Square Roots
We can simplify if 8 contains a Perfect Square
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Simplifying Square Roots
Look to factor perfect squares (4, 9, 16, 25,
36) Put perfect squares in first radical and the
other factor in 2nd. Take square root of first
radical
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Simplifying Square Roots
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Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2,
y6) (Put perfect squares in first radical and
other factor in 2nd)
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Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2,
y6) ( Put perfect squares in first radical and
other factor in 2nd) 2. Take square root of first
radical
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Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2,
y6) ( Put perfect squares in first radical and
other factor in 2nd) 2. Take square root of first
radical
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Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2,
y6) ( Put perfect squares in first radical and
other factor in 2nd) 2. Take square root of first
radical
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Simplifying Square Roots
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Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25,
36) ( Put perfect squares in first radical and
other factor in 2nd)
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Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25,
36) ( Put perfect squares in first radical and
other factor in 2nd) 2. Take square root of first
radical
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Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25,
36) ( Put perfect squares in first radical and
other factor in 2nd)
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Simplifying Radicals
1. Factor any perfect squares (4, 9, 16, 25,
36) ( Put perfect squares in first radical and
other factor in 2nd)
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Simplifying Radicals
1. Factor any perfect squares (4, 9, 16, 25, x2,
y6) ( Put perfect squares in first radical and
other factor in 2nd) 2. Take square root of first
radical
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Simplifying Fraction Radicals
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Simplifying Fraction Radicals
1. Factor any perfect squares (4, 9, 16, 25, x2,
y6) from NUMERATOR DENOM. ( Put perfect
squares in first radical and other factor in 2nd)
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Simplifying Fraction Radicals
1. Factor any perfect squares (4, 9, 16, 25, x2,
y6) from NUMERATOR DENOM. ( Put perfect
squares in first radical and other factor in
2nd) 2. Take square root of first radical
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Simplifying Fraction Radicals
1. Factor any perfect squares (4, 9, 16, 25, x2,
y6) from NUMERATOR DENOM. ( Put perfect
squares in first radical and other factor in
2nd) 2. Take square root of first radical 3.
REDUCE
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Finding Perfect Squares
The most common perfect squares are 4 9. USE
DIVISIBILITY RULES A number is divisible by 4
if the last 2 digits are divisible by 4. ( 23,732
is divisible by 4 since 32 is.) A number is
divisible by 9 if the sum of its digits
are. (4653 is divisible by 9 since the sum of
its digits are.)
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Finding Perfect Squares
MORE EASY TO FIND PERFECT SQUARES A number with
an even number of Zeros. ( 31,300 is divisible by
10, 70,000 is divisible by 100) A number
ending in 25, 50 or 75 is divisible by 25. (425
350 775 are all divisible by 25)
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Perfect Squares Guide Divisibility Rules for
Perfect Squares
4 If Last 2 Digits are Divisible by 4 9 If Sum
of Digits are Divisible by 9 16 Use 4 Rule 25
If ends in 25, 50, 75 or 00 36 Use 4 or 9
Rule 49 Look for multiple of 50 and subtract
multiple 64 Use 4 Rule 81 Use 9 Rule 100 If
ends in 00 121 No Rule, Divide by 121 144 Use
4 or 9 Rule
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Multiply Radicals
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Multiply Radicals

• Multiply to one radical

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Multiply Radicals

• Multiply to one radical

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Multiply Radicals

• Multiply to one radical
• Simplify
• Factor any perfect squares (4, 9, 16, x2,
  y6)

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Multiply Radicals

• Multiply to one radical
• Simplify
• Factor any perfect squares (4, 9, 16, x2,
  y6)

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Multiply Radicals

• Multiply to one radical
• Simplify
• Factor any perfect squares (4, 9, 16, x2,
  y6)
• Take square root of first radical

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Multiply Radicals
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Multiply Radicals

1. Multiply to one radical

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Multiply Radicals

1. Multiply to one radical

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Multiply Radicals

• Multiply to one radical
• Simplify
• Factor any perfect squares (4, 9, 16, x2,
  y6)

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Multiply Radicals

• Multiply to one radical
• Simplify
• Factor any perfect squares (4, 9, 16, x2,
  y6)
• Take square root of first radical

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Multiply Radicals
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Multiply Radicals

1. Multiply to one radical

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Multiply Radicals

1. Multiply to one radical

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Multiply Radicals

• Multiply to one radical
• Simplify
• Factor any perfect squares (4, 9, 16, x2,
  y6)

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Multiply Radicals

• Multiply to one radical
• Simplify
• Factor any perfect squares (4, 9, 16, x2,
  y6)
• Take square root of first radical

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Multiply Radicals

• Multiply to one radical

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Multiply Radicals

• Multiply to one radical

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Multiply Radicals

• Multiply to one radical
• Simplify
• Factor any perfect squares (4, 9, 16, x2,
  y6)

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Multiply Radicals

• Multiply to one radical
• Simplify
• Factor any perfect squares (4, 9, 16, x2,
  y6)
• Take square root of first radical

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Another Way to Multiply Radicals
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Multiply Radicals By Factoring Perfect Squares
First
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Multiply Radicals By Factoring Perfect Squares
First

• Factor any perfect squares

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Multiply Radicals By Factoring Perfect Squares
First

• Factor any perfect squares

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Multiply Radicals By Factoring Perfect Squares
First

• Factor any perfect squares
• Take square roots of perfect squares

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Multiply Radicals By Factoring Perfect Squares
First

• Factor any perfect squares
• Take square roots of perfect squares
• Multiply ,s

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Multiply Radicals with Distributive Prop.
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Multiply Radicals with Distributive Prop.

• Mult. With Distr. Property

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Multiply Radicals with Distributive Prop.

• Mult. With Distr. Property

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Multiply Radicals with Distributive Prop.

• Mult. With Distr. Property
• Factor any perfect squares

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Multiply Radicals with Distributive Prop.

• Mult. With Distr. Property
• Factor any perfect squares
• Take square roots of perfect squares
• Simplify if possible

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Multiply Radicals with Distributive Prop.

• Multiply Terms (Use Multiplying Boxes CLT)
• Simplify if possible

x
3x
3x2
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Dividing Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2,
y6)from the numerator denominator ( Put
perfect squares in first radical and other factor
in 2nd) 2. Take square root of first radical. 3.
Reduce
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Dividing Square Roots
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Dividing Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2,
y6)from the numerator denominator ( Put
perfect squares in first radical and other factor
in 2nd)
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Dividing Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2,
y6)from the numerator denominator ( Put
perfect squares in first radical and other factor
in 2nd)
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Dividing Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2,
y6)from the numerator denominator ( Put
perfect squares in first radical and other factor
in 2nd) 2. Take square root of first radical.
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Dividing Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2,
y6)from the numerator denominator ( Put
perfect squares in first radical and other factor
in 2nd) 2. Take square root of first radical. 3.
Reduce
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Divide Radicals
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Divide Radicals

• Divide

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Divide Radicals

• Divide

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Divide Radicals

• Divide
• Simplify

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Divide Radicals

• Divide
• Simplify

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Divide RadicalsRationalizing the Denominator
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Divide RadicalsRationalizing the Denominator

• Mult. Numerator Denom. By Denom. to get a
  Perfect Square in Denominator

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Divide RadicalsRationalizing the Denominator

• Mult. Numerator Denom. By Denom. to get a
  Perfect Square in Denominator

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Divide RadicalsRationalizing the Denominator

• Mult. Numerator Denom. By Denom. to get a
  Perfect Square in Denominator
• Take the square root of the Perfect Square
• Simplify

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Divide RadicalsRationalizing the Denominator

• Mult. Numerator Denom. By Denom. to get a
  Perfect Square in Denominator
• Take the square root of the Perfect Square
• Simplify

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Divide RadicalsRationalizing the Denominator(2
term)
Rationalizing the denominator means making the
denominator rational or REMOVING IRRATIONAL TERMS
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Divide RadicalsRationalizing the Denominator(2
term)

• Mult. Numerator Denom. By Conjugate to get a
  Perfect Square in Denominator

The conjugate of a binomial is SWITCHING THE
SIGN BETWEEN THEM
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Divide RadicalsRationalizing the Denominator(2
term)

• Mult. Numerator Denom. By Conjugate to get a
  Perfect Square in Denominator

8
8
64
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Divide RadicalsRationalizing the Denominator(2
term)

• Mult. Numerator Denom. By Conjugate to get a
  Perfect Square in Denominator
• Take the square root of the Perfect Square

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Divide RadicalsRationalizing the Denominator(2
term)

• Mult. Numerator Denom. By Conjugate to get a
  Perfect Square in Denominator
• Take the square root of the Perfect Square
• Simplify

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Square Roots of Perfect Squares
1. Factor any perfect squares 2. Take
square root of perfect square.
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Values that make it Real
1. Set term in sq. root 0 2. Solve
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Values that make it Real
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Values that make it Real
1. Set term in sq. root 0 2. Solve
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Values that make it Real
Just note that if x is any real number (all
numbers on the number line) x2 can never be
negative, so (x216) will always be
positive. The answer is any real number
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Negative Square Roots
Can you take the square root of a negative number?
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Can you take the square root of a negative number?
NO
Because a negative number squared is positive
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Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25,
36) ( Put perfect squares in first radical and
other factor in 2nd) 2. Take square root of first
radical
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Simplifying Negative Square Roots
1. Factor any perfect squares (4, 9, 16, 25,
36) ( Put perfect squares in first radical and
other factor in 2nd) 2. Take square root of first
radical 3. The square root of -1 is called
IMAGINARY (We will review this later)
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Adding Radicals
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Adding Radicals
Same as Adding LIKE TERMS
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As Simple As
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As Simple As


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Similar Radical's are Like Terms


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Can't Combine Unlike Terms



a b a b
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Only Combine Like Terms




a a b 2a b
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Always Combine Like Terms


a a 2a
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Always Combine Like Terms


a a 2a
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As Simple As
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As Simple As
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As Simple As
one radical x plus one radical x equals two
radical x's
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Same as Adding LIKE TERMS
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Same as Adding LIKE TERMS
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Same as Adding LIKE TERMS
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Same as Adding LIKE TERMS
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Same as Adding LIKE TERMS
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Same as Adding LIKE TERMS
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Can you add unlike terms?
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NO
You can't combine unlike terms
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Try This One
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Answer
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Write Down Your Answers
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Answers
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Write Down Your Answers
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Answers
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Adding Radicals
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Adding Radicals

• Simplify to Like

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Adding Radicals

• Simplify to Like

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Adding Radicals

1. Simplify to Like
2. Add the Like

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Adding Radicals
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Adding Radicals

1. Simplify to Like GCF

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Adding Radicals

1. Simplify to Like GCF

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Adding Radicals

1. Simplify to Like GCF

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Adding Radicals

1. Simplify to Like GCF
   Simplify

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Adding Radicals

1. Simplify to Like GCF
   Simplify
2. Add the Like Terms

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Adding Radicals
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Adding Radicals

1. Simplify to Like Rationalize Denominator

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Adding Radicals

1. Simplify to Like Rationalize Denominator

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Adding Radicals

1. Simplify to Like Rationalize Denominator
2. Add the Like Terms