Simplifying Radicals: Add & Subtract



Terminology

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Each of these roots stands for a dimension, i.e. 4th root is describing a 4-dimensional object.

Only the 2nd and 3rd roots have special names. 4th, 5th, 6th roots, etc. are named according to their dimension.

Signs:

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Simplify Radicals: Variables

The power of the exponent and the root must be the same number OR the power must be evenly divisible by the root.

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Example:

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If the power is not evenly divisible by the root, you must split the radical into two pieces. The first one will have only those powers that are divisible by the root and the second radical will hold the leftovers.

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Rationalize the Denominator

This process is used on fractions that have radicals ([pic]) in the denominator(bottom). Its purpose is to get rid of the radicals in the denominator.

One method is to multiply the top and bottom by the radical on the bottom.

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Notice that multiplying the bottom radical by itself creates a square root of a square. That leaves the original number (5) without the radical on the bottom.

Examples:

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When the denominator contains a binomial, you must multiply by the complex conjugate in order to cancel out the roots during multiplication.

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Complex conjugate of the denominator: [pic]

[pic] [pic] = [pic]

Distribute the 2x on top and do FOIL on the bottom

[pic] (Notice that the middle terms cancel out)

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Radical Operations: Add & Subtract

Terms must have the same radical in them in order

to add or subtract them.

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These expressions have the same thing under the square root, so we simply add or subtract the coefficients (the numbers in front) and keep the radical.

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If they do not have the same thing under the radical,

check each radical to see if it can be simplified.

[pic] (8 and 2 are not the same, but 8 can be simplified)

Simplify the square root of 8:

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Now the problem becomes:

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Radical Operations: Multiply & Divide

Terms do not need the same radical in them in order

to multiply or divide them.

Rule: Multiply (or Divide) "outside numbers" with "outside numbers"

and "inside numbers" with "inside numbers.

Multiplication:

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distributive property

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FOIL: simplify first, then multiply

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Now that all 4 terms are simplified, we can do FOIL

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There are no like terms, so the last line is the answer

Division with radicals:

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Another example

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The Division is complete, but the radical on the bottom must be rationalized.

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