Section 1: Simplifying Radical Expressions Using the Product Rule and the Quotient Rule

An Overview of Radical Expressions
We begin our discussion in this section by looking at the following: What is 7 squared? The answer, of course, is

72 = 7 · 7 = 49

So when we square 7, we get 49. Now, we want to go in the opposite direction.

The opposite (inverse) of squaring a number is called taking its square root. For example,

1. a square root of 100 is 10 because 10² = 100.
2. another square root of 100 is -10 because (-10)² = 100.

Let c be a real number. If a² = c, then a is a square root of c.

Real numbers have two square roots, one positive and one negative. The positive or principal square root is written with the symbol        and the negative square rootis written with the symbol     . The symbol          is called the radical sign and it always represents the principal square root except that  

A common mistake is to say that   . This is not true. The correct answer is 

The square root of a number is always positive.

The number inside the radical sign is called the radicand. The entire expression is called a radical.

Example 1. Find the square root.

Solution.

Use Product and Quotient Rules for Radicals

When presented with a problem like √4 , we don’t have too much difficulty saying that the answer 2 (since 2 × 2 = 4). Even a problem like ³√ 27 = 3  is easy once we realize 3 × 3 × 3 = 27.

Our trouble usually occurs when we either can’t easily see the answer or if the number under our radical sign is not a perfect square or a perfect cube.

A problem like √24 may look difficult because there is no number that we can multiply by itself to give 24. However, the problem can be simplified. So even though 24 is not a perfect square, it can be broken down into smaller pieces where one of those pieces might be a perfect square. So now we have √24 = √ 4 × 6 = √ 4 · √ 6 = 2√ 6 .

The following rules are very helpful in simplifying radicals.

Rules of Radicals If n is a positive integer greater than 1 and both a and b are positive real numbers then,
1. Inverse Property	n√ an = a if n is odd or n√ an = | a | if n is even
2. Product Rule	n√ ab = n√ a · n√ b
3. Quotient Rule
Note that on occasion we can allow a or b to be negative and still have these properties work.

Also, note that while we can “break up” products and quotients under a radical, we can’t do the same thing for sums or differences.  In other words,

If we “break up” the root into the sum of the two pieces, we clearly get different answers!  So, be careful not to make this very common mistake!

We are going to be simplifying radicals shortly and so we should next define simplified radical form.  A radical is said to be in simplified radical form (or just simplified form) if each of the following are true.

1. All exponents in the radicand must be less than the index.
2. Any exponents in the radicand can have no factors in common with the index.
3. No fractions appear under a radical.
4. No radicals appear in the denominator of a fraction.

Simplifying a radical expression can involve variables as well as numbers. Just as you were able to break down a number into its smaller pieces, you can do the same with variables. When the radical is a square root, you should try to have terms raised to an even power (2, 4, 6, 8, etc).

When the radical is a cube root, you should try to have terms raised to a power of three (3, 6, 9, 12, etc.). For example,  . These types of simplifications with variables will be helpful when doing operations with radical expressions.

Example 2.   Simplify the following radical.

Solution.

 Example 3.      Reduce the radical expression to lowest terms.

Solution.

Example 4.       Simplify the following radical.

  ∛ 320  
Solution.

∛320 = ∛ 64 · 5 = ∛ 64 · ∛ 5 = 4 ∛ 5  

Example 5.     Simplify the following. Assume all variables are positive.

Solution.
In this case the exponent (7) is larger than the index (2) and so the first rule for simplification is violated.  To fix this we will use the first and second properties of radicals above.  So, let’s note that we can write the radicand as follows: 

y7 = y6y = (y3)2y

So, we’ve got the radicand written as a perfect square times a term whose exponent is smaller than the index.  The radical then becomes,

Now use the second property of radicals to break up the radical and then use the first property of radicals on the first term.

This now satisfies the rules for simplification and so we are done.

Before moving on let’s briefly discuss how we figured out how to break up the exponent as we did.  To do this we noted that the index was 2.  We then determined the largest multiple of 2 that is less than 7, the exponent on the radicand.  This is 6.  Next, we noticed that 7 = 6 + 1. 

Finally, remembering several rules of exponents we can rewrite the radicand as,
y⁷ = y⁶y = y⁽³⁾⁽²⁾y = (y³)²y
 

Example 6.  Simplify the following. Assume all variables are positive.

 
Solution.

There is more than one term here but everything works in exactly the same fashion.  We will break the radicand up into perfect squares times terms whose exponents are less than 2 (i.e. 1).

18x⁶y¹¹ = 9x⁶y¹⁰(2y) = 9(x³)²(y⁵)²⁽2y)

Don’t forget to look for perfect squares in the number as well.

Now, go back to the radical and then use the second and first property of radicals as we did in the first example.

Note that we used the fact that the second property can be expanded out to as many terms as we have in the product under the radical.  Also, don’t get excited that there are no x’s under the radical in the final answer.  This will happen on occasions.

Test Your Knowledge by opening up the Test Yourself Activity.