Multiplying and Dividing Radicals

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Multiplying and Dividing Radicals
The product and quotient properties of square
roots can be used to multiply and divide
radicals, because
and
Example 1
Example 2
Example 3
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Product Rule

• Simplify radicals
• Multiply Coefficients
• Multiply radicands
• Roots must be the same
• Simplify, if needed

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Examples Product Rule
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Quotient Rule

• Fractions made up of radicals can be simplified
  just like fractions

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5
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1. Simplify.

• Multiply the radicals.

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6. Simplify.

• Multiply the coefficients and the radicals.

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7. Simplify.

• Divide the radicals.

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8. Simplify.
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Rationalizing Radicals
To simplify a fraction with a radical in the
denominator, multiply the numerator and
denominator by the radical.
Example 1
Estimation is easier with rational denominators.
This process is called rationalizing the
denominator.
Example 2
Since the square root of a quotient is a quotient
of square roots, the square root of a fraction
must be rationalized to be in simplest form.
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9. Simplify.

• Answer

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Adding and Subtracting Radicals
Radicals that represent the square root of the
same number can be treated as a common factor.
Examples
Radicals representing square roots of different
numbers can not be gathered like this.
But
simplifying sometimes results in multiples of the
same radical, which can be.
Examples
Like terms can be gathered. Unlike terms can not.
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Combining Like Terms

• Radicals Like Terms
• Same variables
• Variables have the same exponents
• IDENTICAL RADICALS
• Examples

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• Simplify radicals if possible
• Combine coefficients

Radicals ARE simplified
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1. Simplify.

• Just like when adding variables, you can only
  combine LIKE radicals.
• 5 v5

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2. Simplify.

• Answer 4 v7 3 v3

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3. Simplify.

• Simplify each radical.
• 4v93 - 2v16 3 2v45
• 4 3v3 - 2 4v32 2v5
• 12v3 - 8v3 4v5
• Combine like radicals
• 4v3 4v5

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More Radical Fun
SIMPLIFY
Must have Common Denominators
MULTIPLY
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Distributive Property with Radicals
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Multiplying Binomials With Radicals
Multiplying binomials that contain radicals
sometimes results in products of radicals that
can be simplified.
Examples
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Conjugates
Multiplying conjugates like these together
results in a rational number
Conjugates are therefore used to rationalize
certain fractions.
Example
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Practice
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