This lesson is going to take a "quick look" at adding, subtracting, multiplying and dividing square roots. For more information regarding these operations, refer to the Radical Section under Algebra 1.

Add (and Subtract):

 


Adding or subtracting radicals is the same concept as that of adding or subtracting similar, or "like", terms. The index and the values under the radical (the radicands) must be the SAME (creating "like radicals") before you can add or subtract the radical expressions.

statement
Adding and subtracting radicals: For radicals having the same index and the same values under the radical (the radicands), add (or subtract) the values directly in front of the radicals and keep the radical.

You can think of adding radicals like adding 4x + 6x which is 10x.
You have 4 x's and 6 more x's, so you have 10 x's all totaled.

ADDSUBTRACT

When radicals are the SAME, add the numbers right tight in front of the radicals.
Think: there are 4 square roots of 3 plus 6 square roots of 3.
That makes 10 square roots of 3 in total.

It may be necessary to simplify first!
Different looking radicals may actually be the same when simplified.
bulletrad add2
At first glance, it appears that combining these terms under addition is not possible since the radicals are not the same. But if we look further, we can simplify the second term so it will be a "like" radical:
rad a 2
ANSWER: rad2b


Some radicals cannot be added!
bulletrad3
The radicals are different and each is already in simplest form. There is simply no way to combine these values. The answer is the same as the original problem.
ANSWER: rad 2 c


Remember, there may be an unwritten ONE in front of a radical!
bullet rad as4
There is an implied "1" in front of rad3 as. All radicals are already in simplest form. Combine the "like" radicals.
rad SA4 aa
ANSWER: rad as4 ans

bullet REMEMBER: Always simplify first! When the radicals in an addition or subtraction problem are different, be sure to check to see if the radicals can be simplified. It may be the case that when the radicals are simplified, they will become "like" radicals, making it possible for them to be added or subtracted.


Multiply:

 

 

statement
Multiplying Radicals: When multiplying radicals (with the same index), multiply under the radical, and then multiply in front of the radical (any values multiplied times the radicals).

multiply

Multiply out front and multiply under the radicals.

It may be necessary to simplify the answer!
Different looking radicals may actually be the same when simplified.
bullet mu2
Multiply out front and multiply under the radicals:
mu22
Then simplify the result.
mu222
ANSWER: mu2222


Divide:

 

 

statement
Dividing Radicals: When dividing radicals (with the same index), divide under the radical, and then divide in front of the radical (divide any values multiplied times the radicals).

divide
Divide out front and divide under the radicals.

Radicals in the denominator of a fraction:
bullet d2
This fraction will be in simplified form when the radical is removed from the denominator.

You need to create a perfect square under the square root radical in the denominator by multiplying the top and bottom of the fraction by the same value (this is actually multiplying by "1"). The easiest approach is to multiply by the square root radical you need to convert (in this case multiply by d1).
d3
ANSWER: d4 This is now in simplified form.

This process of simplifying is called "rationalizing the denominator" as it creates a fraction with an integer denominator.

 

divider

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