# Algebra II : Understanding Radicals

## Example Questions

### Example Question #51 : Understanding Radicals

Rationalize the denominator and simplify:

Explanation:

To rationalize a denominator, multiply all terms by the conjugate. In this case, the denominator is , so its conjugate will be .

So we multiply: .

After simplifying, we get .

### Example Question #1 : Radicals As Exponents

Which of the following answer choices best simplifies ?

Explanation:

The first step to simplifying a problem like this one is to convert all radicals to fractional exponents. Remember the following relationship:

Also keep in mind your exponent rules, especially this one:

Now, let's get started on this problem. First, we change that radical expression into something with fractional exponents instead.

Now we use our exponent rules to simplify the numerator.

Finally, we simplify the entire fraction:

We can leave it like this, but it would be better to write it this way, without negative exponents:

### Example Question #2 : Radicals As Exponents

Simplify:

You may assume that the radicand is nonnegative.

Explanation:

The polynomial , being a polynomial of degree 2, cannot be a cube of another polynomial. Also, it does not fit the pattern of a perfect square binomial, since its constant term is negative. Therefore, we cannot extract a root of the polynomial to help to simplify it.

We can, however, rewrite each root as a fractional exponent, apply the power of a power property, then convert back, as follows:

### Example Question #3 : Radicals As Exponents

Simplify:

You may assume that  is a nonnegative real number.

Explanation:

The best way to simplify a radical within a radical is to rewrite each root as a fractional exponent, then convert back.

First, rewrite the roots as exponents.

We can simplify this further:

### Example Question #4 : Radicals As Exponents

What is the value of ?

Explanation:

Recall that when you have a fractional exponent, this means that you have a root involved.  The denominator of the exponent is the type of root.  Our question's exponent is:

Therefore, the root is  or a square root.

The numerator is the power for the base.  Therefore, we can rewrite our problem as:

Now, to simplify this, we could do:

Using our exponent rules, this is:

Factoring out  sets of , we get:

, or

### Example Question #5 : Radicals As Exponents

What is the value of ?

Explanation:

Recall that when you have a fractional exponent, this means that you have a root involved.  The denominator of the exponent is the type of root.  Our question's exponent is:

Therefore, the root is  or a cube root.

The numerator is the power for the base.  Therefore, we can rewrite our problem as:

Now, to simplify this, we could do:

Using our exponent rules, this is:

Factoring out  sets of  and  set of , we get:

### Example Question #6 : Radicals As Exponents

What is the value of ?

Explanation:

Recall that when you have a fractional exponent, this means that you have a root involved.  The denominator of the exponent is the type of root.  Our question's exponent is:

Therefore, the root is  or a cube root.

The numerator is the power for the base.  Therefore, we can rewrite our problem as:

Now, to simplify this, we could do:

We can factor out a set of .  This leaves us with:

Simplifying, this is:

### Example Question #7 : Radicals As Exponents

Which of the following is equivalent to

Explanation:

Recall that the cube root of a number is the number that when multiplied by itself 3 times, yields your number.

Thus, we want y, where:

Consider

That cannot be our solution.

Then try

Thus, this must be our solution.

Next time, remember that radicals can be represented by fractional exponents!

### Example Question #8 : Radicals As Exponents

Choose the best answer.  Reduce the following in exponential form:

Explanation:

Simplify the inner term within the parentheses.

### Example Question #9 : Radicals As Exponents

Simplify,

can't be simplified

Explanation:

First write the square root of  as an exponent,

From the rules of exponents we know we can simplyfy by adding the exponents,