Introduction
This set of practice questions will give you practice in operating with radicals. You will not always be able to factor polynomials by factoring whole numbers and whole number coefficients. Nor do all trinomials with whole numbers have whole numbers for solutions. In these last chapters, you will need to know how to operate with radicals.
The radical sign tells you to find the root of a number. The number under the radical sign is called the radicand. Generally, a number has two roots, one positive and one negative. It is understood in mathematics that or is telling you to find the positive root. The symbol tells you to find the negative root. The symbol asks for both roots.
Tips for Simplifying Radicals
Simplify radicals by completely factoring the radicand and taking out the square root. The most thorough method for factoring is to do a prime factorization of the radicand. Then you look for square roots that can be factored out of the radicand.
e.g.,
or
You may also recognize perfect squares within the radicand. Then you can simplify their roots out of the radical sign.
It is improper form for the radicand to be a fraction. If you get rid of the denominator within the radical sign, you will no longer have a fractional radicand. This is known as rationalizing the denominator.
e.g.,
When there is a radical in the denominator, you can rationalize the expression as follows:
If the radicands are the same, radicals can be added and subtracted as if the radicals were variables.
e.g.,
Product property of radicals
When multiplying radicals, multiply the terms in front of the radicals, then multiply the radicands and put that result under the radical sign.
e.g.,
Quotient property of radicals
When dividing radicals, first divide the terms in front of the radicals, and then divide the radicands.
e.g.,
Practice Questions
Simplify the following radical expressions.
Answers
Numerical expressions in parentheses like this [ ] are operations performed on only part of the original expression. The operations performed within these symbols are intended to show how to evaluate the various terms that make up the entire expression.
Expressions with parentheses that look like this ( ) contain either numerical substitutions or expressions that are part of a numerical expression. Once a single number appears within these parentheses, the parentheses are no longer needed and need not be used the next time the entire expression is written.
When two pair of parentheses appear side by side like this ( )( ), it means that the expressions within are to be multiplied.
Sometimes parentheses appear within other parentheses in numerical or algebraic expressions. Regardless of what symbol is used, ( ), { }, or [ ], perform operations in the innermost parentheses first and work outward.
Underlined expressions show simplified result.
1. First, factor the radicand. 

Now take out the square root of any pair of factors or any perfect squares you recognize. 

2. First, factor the radicand. 

Now take out the square root. 

3. First, factor the radicand. 

Now take out the square root. 

4. First, factor the radicand and look for squares. 

Now take out the square root. 

5. Although it looks complex, you can still begin by factoring the terms in the radical sign. 

Factoring out the squares leaves 

This result can be written a few different ways. 

6. First, factor the radicand and look for squares. 

Take out the square roots. 

7. Use the quotient property of radicals and rationalize the denominator. 

8. For this expression, you must rationalize the denominator. Use the identity property of multiplication and multiply the expression by 1 in a form useful for your purposes. In this case, that is to get the radical out of the denominator. 

9. First, rationalize the denominator for this expression. Then see if it can be simplified any further. As in the previous problem, multiply the expression by 1 in a form suitable for this purpose. 

Use the product property of radicals to combine the radicands in the numerator. In the denominator, a square root times itself is the radicand by itself. 

Now factor the radicand. 

Factoring out the square roots results in 

The x in the numerator and the denominator divides out, leaving 

10. In this expression, add the "like terms'' as if the similar radicals were similar variables. 

The radical terms cannot be added further because the radicands are different. 
11. Simplify the second term of the expression by factoring the radicand. 

Now simplify the radicand. 

Finally, combine like terms. 

12. For this expression, use the product property of radicals and combine the factors in the radicand and outside the radical signs. 

13. Use the product property of radicals to simplify the expression. 

Now look for a perfect square in the radicand. You can multiply and then factor or just factor first. 

Or if you just factor the radicand, you will see the perfect square as 5 times 5. 

14. You can start by using the product property to simplify the expression. 

Now use the quotient property to simplify the first radical. 

You should recognize the perfect squares 9 and 16. Simplify the fraction and factor the radicand of the second radical term. 

15. Use the quotient property to get the denominator out of the radicand. 

Now simplify the radicand by factoring so that any perfect squares will come out of the radical sign. 

The common factor in both the numerator and denominator divides out, and you are left with 

16. For this term, factor the radicand in the numerator and look for perfect squares. 

Use the product property in the numerator. 

Divide out the common factor in the numerator and the denominator. 

17. You could proceed in the same way as the previous solution, but let's try another way. Rationalize the denominator. 

Use the product property to simplify the numerator. 

Factor the radicand seeking perfect squares. 

Simplify the numerator. 

Divide out the common factor in the numerator and denominator and you're done. 

18. You use the quotient property to begin rationalizing the denominator. 

The 5 becomes part of the numerator. Factor the numerator and simplify the perfect square in the denominator. 

Simplify the numerator. 

Factor 2 out of the numerator and denominator. 

19. Begin simplifying this term by rationalizing the denominator. 

Using the product property, simplify the numerator and write the product of the term in the denominator. 

Divide out the common factor in the numerator and denominator. Then factor the radicand looking for perfect squares. 

20. You will have to rationalize the denominator, but first factor the radicands and look for perfect squares. 

You can simplify the whole numbers in the numerator and denominator by a factor of 3. 

Now rationalize the denominator. 

Simplify terms by using the product property in the numerator and multiplying terms in the denominator. 

21. Factor the radicand in the last radical. 

Simplify the perfect square. 

Use the commutative property of multiplication. 

Now simplify terms. 

22. Using the product property, put all terms in one radical sign. 

Now use the quotient property to continue simplifying. 

Simplify the perfect squares in the numerator and denominator. 

The expression is fine the way it is, or it could be written as 

23. Begin by factoring the radicand of the second radical. 

Now use the product property to separate the factors of the second radical term into two radical terms. Why? Because you will then have the product of two identical radicals. 

Now simplify the whole numbers. 

24. There is more than one way to simplify an expression. Start this one by using the product property to combine the radicands. 

Now factor the terms in the radicand and look for perfect squares. 

Simplify the radical. 

Another way is to multiply 6 and 54 to get 324, another perfect square. Then 3 times 18 equals 54. 
25. Begin by rationalizing the denominator. 

Use the product property to simplify the numerator. 

Simplify the whole numbers in the numerator and denominator. Then factor the radicand and look for perfect squares. 

Simplify the numerator and divide out common factors in the numerator and denominator. 
