Multiplying and Dividing Radicals - Algebra II

When multiplying radicals, simply multiply the numbers inside the radical with each other. Therefore:

We cannot further simplify because both of the numbers multiplied with each other were prime numbers.

We can solve this by simplifying the radicals first:

Plugging this into the equation gives us:

When dealing with radicals in the denominator, we simplify it by multiplying top and bottom by the radical.

To solve this expression, multiply the numerator and the denominator by the complex conjugate of the denominator. Since the denominator is , the complex conjugate of this is . Therefore:

FOIL with difference of squares. The multiplying cancels the square roots on both terms.

To simplify, you must use the Law of Exponents.

First you must multiply the coefficients then add the exponents:

.

Note: the product of the radicals is the same as the radical of the product:

which is

Once we understand this, we can plug it into the equation:

We can simplify the radicals:

and

Plug in the simplifed radicals into the equation:

We can only simplify the radical in the numerator:

Plugging in the simplifed radical into the equation we get:

Note: We simplified further because both the numerator and denominator had a "4" which canceled out.

Now we want to rationalize the denominator,

First, simplify to .

Then set up the multiplication problem:

.

Multiply the terms outside of the radical, then the terms under the radical:

then simplfy:

The radical is still not in its simplest form and must be reduced further:

. This is the radical in its simplest form.

To divide the radicals, simply divide the numbers under the radical and leave them under the radical:

Then simplify this radical:

.

When multiplying radicals, just take the values inside the radicand and perfom the operation.

can't be reduced so this is the final answer.

When multiplying radicals, just take the values inside the radicand and perfom the operation.

In this case, we have a perfect square so simplify that first.

Then, take that answer and multiply that with to get the final answer.

.

When dividing radicals, check the denominator to make sure it can be simplified or that there is a radical present that needs to be fixed. Since there is a radical present, we need to eliminate that radical. To do this, we multiply both top and bottom by . The reason is because we want a whole number in the denominator and multiplying by itself will achieve that. By multiplying itself, it creates a square number which can be reduced to .

With the denominator being , the numerator is . Final answer is .

When dividing radicals, check the denominator to make sure it can be simplified or that there is a radical present that needs to be fixed.

Both and are perfect squares so they can be simplify.

Final answer is

.

Divide the numbers inside the radicals.

When multiplying radicals, just take the values inside the radicand and perfom the operation.

Since there is a number outside of the radicand, multiply the outside numbers and then the radicand.

When multiplying radicals, just take the values inside the radicand and perfom the operation.

Since there is a number outside of the radicand, multiply the outside numbers and then the radicand.

Before we say that's the final answer, check the radicand to see that there are no square numbers that can be factored. A can be factored and thats a perfect square. When I divide with , I get which doesn't have perfect square factors.

Therefore, our answer becomes

.

When multiplying radicals, just take the values inside the radicand and perfom the operation.

Since there is a number outside of the radicand, multiply the outside numbers and then the radicand.