Algebra II : Factoring Radicals

Study concepts, example questions & explanations for Algebra II

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Example Questions

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Example Question #1 : Factoring Radicals

Simplify the expression:

Possible Answers:

.

Correct answer:

Explanation:

Use the multiplication property of radicals to split the fourth roots as follows:

Simplify the new roots:

Example Question #2 : Factoring Radicals

Simplify the expression.

Possible Answers:

Correct answer:

Explanation:

Use the multiplication property of radicals to split the perfect squares as follows:

Simplify roots,

 

Example Question #3 : Factoring Radicals

Simplify the radical .

Possible Answers:

None of the other answers

Correct answer:

Explanation:

To simplify radicals, we need to factor the expression inside the radical. A radical can only be simplified if one of the factors has a square root that is an integer.

For this problem, we'll first find all of the possible radicals of 12: 1 & 12, 2 & 6, and 3 & 4. Then we look at each factor and determine if any of them has a square root that is an integer. The only one that does is 4, which has a square root of 2. We can rewrite the radical as  which can also be written as . Taking the squareroot of 4, we come to the answer: .

Example Question #4 : Factoring Radicals

Simplify the following radical by factoring the radicand:

Possible Answers:

Correct answer:

Explanation:

The goal of simplifying a radical by factoring is to find a factor of the radicand that has a neat whole number as a square root. If this factor is difficult to determine simply by looking at the radicand, a good way to start is by factoring the radicand until you notice a factor that has a whole number as a square root, and can therefore be taken out of the radical. Assuming we couldn't identify a factor of 150 with a neat square root, we could start the factoring by taking out the smallest factor possible, which for even numbers would be 2:

Neither number has a neat square root, so we'll continue by factoring out the next smallest factor of 3 from 75:

At this point we can see that one of our factors, 25, has a neat square root of 5, which we can take out from under the radical, and now that none of the factors left under the radical can be simplified any further, we simply multiply them back together to give us the most simplified form of our radical:

Example Question #5 : Simplifying Radicals

Simplify the following expression involving radicals by factoring the radicands:

Possible Answers:

Correct answer:

Explanation:

In order to simplify each radical, we must find the factors of its radicand that have a whole number as a square root, which will allow us to take the square root of that factor out of the radical. We start by factoring each radicand, looking for any factors that have a neat whole number as a square root:

After factoring each radicand, we can see that there is a perfect square in each: 25 in the first, 49 in the second, and 4 in the third. Because these factors are perfect squares, we can easily take their square root out of the radical, which then gets multiplied by the coefficient already in front of the radical:

After simplifying each radical, we're left with the same value of in each term, so we can now add all of our like terms together to completely simplify the expression:

Example Question #6 : Simplifying Radicals

Simplify the radical expression.

Possible Answers:

Correct answer:

Explanation:

In order to solve this equation, we must see how many perfect cubes we can simplify in each radical.

First, let's simplify the coefficient under the radical.  is the perfect cube of . Therefore, we can remove  from under the radical, and what we have instead is:

Now, in order to remove variables from underneath the square root symbol, we need to remove the variables by the cube. Since radicals have the property

we can see that

With the expression in this form, it is much easier to see that we can remove one cube from , two cubes from , and two cubes from , and therefore our solution is:

Example Question #5 : Factoring Radicals

Possible Answers:

Correct answer:

Explanation:
  =    =  

=
=

Example Question #6 : Factoring Radicals

Simplify the radical.

Possible Answers:

Cannot be simplified further.

Correct answer:

Explanation:

Find the factors of 128 to simplify the term.

We can rewrite the expression as the square roots of these factors.

Simplify.

Example Question #93 : Real Numbers

Simplify the radical.

Possible Answers:

Correct answer:

Explanation:

Start by finding factors for the radical term.

We can rewrite the radical using these factors.

Simplify the first term.

Example Question #7 : Factoring Radicals

Simplify.

Possible Answers:

Correct answer:

Explanation:

Always work the math under the radical before simplifying. We can't do any math so let's see if it's factorable. This isn't factorable either so the answer is just the problem stated. 

 If you don't believe it, let  and 

 

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