Simplifying Square Root Expressions

1
Simplifying Square Root Expressions
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Numbers with a Root

• Radical numbers are typically irrational numbers
  (unless they simplify to a rational number). Our
  calculator gives

But the decimal will go on forever and not repeat
because it is an irrational number. For the
exact answer just use
Some radicals can be simplified similar to
simplifying a fraction.
3
Radical Product Property

• ONLY when a0 and b0
• For Example

Equal
4
Perfect Squares

• The square of whole numbers.

1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 ,
121, 144 , 169 , 196 , 225, etc
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Simplifying Square Roots

• Check if the square root is a whole number
• Find the biggest perfect square (4, 9, 16, 25,
  36, 49, 64) that divides the number in the root
• Rewrite the number in the root as a product
• Simplify by taking the square root of the perfect
  square and putting it outside the root
• CHECK!
• Note A square root can not be simplified if
  there is no perfect square that divides it. Just
  leave it alone.
• ex v15 , v21, and v17

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Simplifying Square Roots

• Write the following as a radical (square root) in
  simplest form

Simplify.
36 is the biggest perfect square that divides 72.
Rewrite the square root as a product of roots.
Ignore the 5 multiplication until the end.
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Simplifying Square Roots

• Simplify these radicals

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Adding and Subtracting Radicals

• Simplify the expressions

Always simplify a radical first.
Treat the square roots as variables, then
combine like terms ONLY.
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Multiplication and Radicals

• Simplify the expression

Use the Commutative Property to Rewrite the
expression.
Simplify and use the Radical Product Property
Backwards.
If possible, simplify more.
Conclusion Multiply the numbers outside of the
square root, then multiply the numbers inside of
the square root. Then simplify.
10
Distribution and Radicals

• Rewrite the expression

3v6 -2v3
Find the Sum.
-10v18
15v36
5v6 4v3
-30v2
90
12v18
-8v9
36v2
-24
Combine like terms.
Remember Multiply the numbers outside of the
square root, then multiply the numbers inside of
the square root. Then simplify.
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Fractions and Radicals

• Simplify the expressions

There is nothing to simplify because the square
root is simplified and every term in the fraction
can not be divided by 10.
Make sure to simplify the fraction.
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Radical Quotient Property

• ONLY when a0 and b0
• For Example

Equal
13
The Square Root of a Fraction

• Write the following as a radical (square root) in
  simplest form

Take the square root of the numerator and the
denominator
Simplify.
14
Rationalizing a Denominator

• The denominator of a fraction can not contain a
  radical. To rationalize the denominator
  (rewriting a fraction so the bottom is a rational
  number) multiply by the same radical.
• Simplify the following expressions

15
WARNING

• In general

For Example
Not Equal