# Algebra II : Understanding Radicals

## Example Questions

### Example Question #21 : Understanding Radicals

Evaluate:

Explanation:

Evaluate each square root.  The square root of a number evaluates into a number which multiplies by itself to achieve the number in the square root.

Substitute the terms back into the expression.

### Example Question #22 : Understanding Radicals

Solve:

Explanation:

Solve each radical.  The square root determines a number that multiples by itself to equal the number inside the square root.

Rewrite the expression.

### Example Question #21 : Radicals

True or false:  is a radical expression in simplest form.

False

True

False

Explanation:

A radical expression which is the th root of a constant is in simplest form if and only if, when the radicand is expressed as the product of prime factors, no factor appears  or more times.  Since  is a square, or second, root, find the prime factorization of 52, and determine whether any prime factor appears two or more times.

52 can be broken down as

and further as

The factor 2 appears twice, so  is not in simplest form.

### Example Question #21 : Understanding Radicals

Explanation:

To solve this, remember that when multiplying variables, exponents are added.  When raising a power to a power, exponents are multiplied.  Thus:

### Example Question #2 : Non Square Radicals

Simplify by rationalizing the denominator:

Explanation:

Since , we can multiply 18 by  to yield the lowest possible perfect cube:

Therefore, to rationalize the denominator, we multiply both nuerator and denominator by  as follows:

### Example Question #3 : Non Square Radicals

Simplify:

Explanation:

Begin by getting a prime factor form of the contents of your root.

Applying some exponent rules makes this even faster:

Put this back into your problem:

Now, we can factor out  sets of  and  set of .  This gives us:

### Example Question #4 : Non Square Radicals

Simplify:

Explanation:

Begin by factoring the contents of the radical:

This gives you:

You can take out  group of .  That gives you:

Using fractional exponents, we can rewrite this:

Thus, we can reduce it to:

Or:

### Example Question #22 : Understanding Radicals

Simplify:

Explanation:

To simplify , find the common factors of both radicals.

### Example Question #6 : Non Square Radicals

Simplify:

Explanation:

To take the cube root of the term on the inside of the radical, it is best to start by factoring the inside:

Now, we can identify three terms on the inside that are cubes:

We simply take the cube root of these terms and bring them outside of the radical, leaving what cannot be cubed on the inside of the radical.

Rewritten, this becomes