# Algebra II : Understanding Radicals

## Example Questions

### Example Question #41 : Understanding Radicals

Simplify:

Explanation:

Multiply the integers and combine the radicals together by multiplication.

Break up square root of 800 by common factors of perfect squares.

### Example Question #42 : Understanding Radicals

Evaluate:

Explanation:

Multiply the integers and the value of the square roots to combine as one radical.

Simplify the radical.  Use factors of perfect squares to simplify root 300.

### Example Question #43 : Understanding Radicals

Simplify:

Explanation:

In order to simplify the radical, we will need to pull out common factors of possible perfect squares.

The expression becomes:

The radical 14 does not have any common factors of perfect squares.

### Example Question #44 : Understanding Radicals

Simplify:

Explanation:

Simplify by factoring both radicals by perfect squares.

Replace the terms.

Combine like-terms.

### Example Question #45 : Understanding Radicals

Solve:

Explanation:

Multiply the integers outside of the radical.

Multiply all the values inside the radicals to combine as one radical.

Rewrite the radical using factors of perfect squares.

### Example Question #46 : Understanding Radicals

Simplify:

Explanation:

The values of the radicals can be combined my multiplication.

The value of  can be factored by using known perfect squares as factors.

Replace the term.

### Example Question #47 : Understanding Radicals

Simplify:

Explanation:

Rewrite each radical as common factors using known perfect squares.

Simplify the expression.

### Example Question #48 : Understanding Radicals

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

True

False

True

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 cube, or third, root, find the prime factorization of 52, and determine whether any prime factor appears three or more times.

52 can be broken down as

and further as

No prime factor appears three or more times, so  is in simplest form.

### Example Question #49 : Understanding Radicals

Try without a calculator.

Simplify:

Explanation:

First, apply the Quotient of Radicals Property to split the radical into a numerator and a denominator:

Since we are dealing with cube, or third, roots, rationalize the denominator by multiplying both halves of the fraction by the least cube-root radical expression that would eliminate the radical in the denominator. To do this, note that 7 is a prime number. Therefore, to get a perfect cube, it is necessary to multiply both halves by , and apply the Product of Radicals Property. The reason for this is made more apparent below:

### Example Question #50 : Understanding Radicals

Try without a calculator.

Simplify: