## Example Questions

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### Example Question #1 : Radicals

Subtract and simplify:

Explanation:

Find the lease common denominator:

A radical cannot be in the denominator:

Find the value of .

Explanation:

To solve this equation, we have to factor our radicals.  We do this by finding numbers that multiply to give us the number within the radical.

4 is a perfect square, so we can find the root:

Since both have the same radical, we can combine them:

### Example Question #2 : Radicals

Simplify the following equation:

Cannot simplify further

Explanation:

When simplifying, you should always be on the lookout for like terms. While it might not look like there are like terms in , there are -- we just have to be able to rewrite it to see.

Before we start combining terms, though, let's look a little more closely at this part:

We need to "distribute" that exponent to everything in the parentheses, like so:

But 4 to the one-half power is just the square root of 4, or 2.

Okay, now let's see our equation.

We need to start combining like terms. Take the terms that include x to the one-half power first.

Now take the terms that have x to the one-third power.

All that's left is to write them in order of descending exponents, then convert the fractional exponents into radicals (since that's what our answer choices look like).

### Example Question #3 : Radicals

Solve.

Explanation:

When adding and subtracting radicals, make the sure radicand or inside the square root are the same.

If they are the same, just add the numbers in front of the radical. If they are not the same, the answer is just the problem stated.

Since they are the same, just add the numbers in front of the radical:  which is

Therefore, our final answer is the sum of the integers and the radical:

### Example Question #4 : Radicals

Solve.

Explanation:

When adding and subtracting radicals, make the sure radicand or inside the square root are the same.

If they are the same, just add the numbers in front of the radical.

If they are not the same, the answer is just the problem stated.

Since they are the same, just add and subtract the numbers in front:  which is

### Example Question #5 : Radicals

Simplify, if possible:

Explanation:

The radicals given are not in like-terms.  To simplify, take the common factors for each of the radicals and separate the radicals.  A radical times itself will eliminate the square root sign.

Now that each radical is in its like term, we can now combine like-terms.

### Example Question #6 : Radicals

Multiply and express the answer in the simplest form:

Explanation:

### Example Question #7 : Radicals

Simplify.

Explanation:

We can solve this by simplifying the radicals first:

Plugging this into the equation gives us:

### Example Question #8 : Radicals

What is the product of  and ?

Explanation:

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.

### Example Question #9 : Radicals

Simplify the following: