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Related Lessons: Adding and Subtracting Radical Expressions Multiplying Radical Expressions Rationalizing the Denominator

Simplifying Radical Expressions | Step by Step

 

A radical expression is composed of three parts: a radical symbol, a radicand and an index. In this tutorial, the primary focus is on simplifying radical expressions with index of 2. This type of radical is commonly known as the square root.

If you want to skip over the introductory discussion, you may jump ahead into the fifteen (14) worked examples.

Components of a Radical Expression

 

diagram showing parts of a radical expression: index, radical symbol, and radicand

 

Starting with a single radical expression, we want to break it down into pieces of "smaller" radical expressions. We hope that some of those pieces can be further simplified because the radicands (stuff inside the symbol) are perfect squares. Remember, the square root of perfect squares comes out very nicely!

We need to recognize how a perfect square number or expression may look like. Let's do that by going over concrete examples.

Examples of Perfect Squares
Numbers Variables
4 x2 
9 y4 
16 m6 
25 k12 
36 a2 b4  
49 u18 v8 w22 

 

What makes them perfect squares?

  • Notice that the square root of each number above yields a whole number answer. Think of them as perfectly well behaved numbers.
examples of the square root of perfect square numbers:√4=2, √9=3, √16=4, √25=5, √36=6 and √49=7
  • In addition, those numbers are perfect squares because they all can be expressed as exponential numbers with even powers. The powers don't need to be "2" all the time. As long as the powers are even numbers such 2, 4, 6, 8, etc, they are considered to be perfect squares. One way to think about it, a pair of any number is a perfect square!
√(2^2)=2, √(3^2)=3, √(4^2)=4, √(5^2)=5, √(6^2)=6, √(7^2)=7
  • More so, the variable expressions above are also perfect squares because all variables have even exponents or powers.

 


try this out! WORKED EXAMPLES

Example 1: Simplify the radical expression square root of 16 See solution
Example 2: Simplify the radical expression square root of 60 See solution
Example 3: Simplify the radical expression square root of 72 See solution
Example 4: Simplify the radical expression square root of 48 See solution
Example 5: Simplify the radical expression square root of 200 See solution
Example 6: Simplify the radical expression square root of 180 See solution
Example 7: Simplify the radical expression square root of 12 x^2y^4 See solution
Example 8: Simplify the radical expression square root of 54a^10b^16c^7 See solution
Example 9: Simplify the radical expression square root of 400 h^3k^9m^7n^13 See solution
Example 10: Simplify the radical expression square root of 147*w^6*q^7*r^27 See solution
Example 11: Simplify the radical expression square root of 32 You Try!
Example 12: Simplify the radical expression square root of 125 You Try!
Example 13: Simplify the radical expression square root of 80x^3yz^5 You Try!
Example 14: Simplify the radical expression √(18*m^11*n^12*k^13) You Try!

Example 1: Simplify the radical expression square root of 16.

This is an easy one! The number 16 is obviously a perfect square because I can find a whole number that when multiplied by itself gives the target number. It must be 4 since (4)(4) =  42 = 16. Thus, the answer is √(16)=4.

Below is a screenshot of the answer from the calculator which verifies our answer.

shows the answer for √(16)=4 using TI calculator

 


Example 2: Simplify the radical expression square root of 60.

You can do some trial and error to find a number when squared gives 60. Going through some of the squares of the natural numbers...

There is no whole number when multiplied gives the value of 60.

The answer must be some number n found between 7 and 8. So we expect that the square root of 60 must contain decimal values. To simplify this radical number, try factoring it out such that one of the factors is a perfect square. By quick inspection, the number 4 is a perfect square that can divide 60.

So our answer is...

√60=2*√15

And for our calculator check...

verifies answer of √60=2*√15 using TI calculator

 

What rule did I use to break them as product of square roots? Here it is!

Product Rule of Square Roots

√(ab)=√a*√b where a>0 and b>0

  • This means, the square root of the product of a and b 

is equal to the product of their individual square roots.

warning symbol Remember this simple "formula". You will use this over and over again.

 


Example 3: Simplify the radical expression square root of 72.

Always look for a perfect square factor of the radicand. After doing some trial and error, I found out that any of the perfect squares 4, 9 and 36 can divide 72. So which one should I pick? Actually, any of the three perfect square factors should work. However, the best option is the largest possible one because this greatly reduces the number of steps in the solution.

Compare what happens if I simplify the radical expression using each of the three possible perfect square factors. Picking the largest one makes the solution very short and to the point.

Start with 4 Start with 9 Start with 36
√72=2*3*√2=6√2 √72=√9*√8=6√2 √36=√36*√2=6*√2

it's okay if ever you start with the smaller perfect square factors. You just need to make sure that you further simplify the leftover radicand (stuff inside the radical symbol). Repeat the process until such time when the radicand no longer has a perfect square factor.

Looks like the calculator agrees with our answer. Great!

verifies the answer using TI calculator of √72=6*&radic2

 

 


Example 4: Simplify the radical expression square root of 48.

Another way to solve this is to perform prime factorization on the radicand. Then express the prime numbers in pairs as much as possible. The paired prime numbers will get out of the square root symbol, while the single prime will stay inside.

√48=4*√3

And it checks when solved in the calculator.

verifies answer with TI 84 calculator √48=4√3

 


Example 5: Simplify the radical expression √200.

Let's find a perfect square factor for the radicand. Although 25 can divide 200, the largest one is 100. Next, express the radicand as products of square roots, and simplify.

√200=10*√2

The calculator gives us the same result!

answer using calculator √200=10*√2

 

 


Example 6: Simplify the radical expression √180.

smiley face  Work this out on paper first, and click here to compare answer.

 

 


Example 7: Simplify the radical expression √(12*x^2*y^4).

The radicand contains both numbers and variables. Let's deal with them separately.

For the numerical term 12, its largest perfect square factor is 4. Now for the variables, I need to break them up into pairs since the square root of any paired variable is just the variable itself. The solution to this problem should look something like this...

√(12*x^2*y^4)=2xy^2√3

tip The standard way of writing the final answer is to place all the terms (both numbers and variables) that are outside the radical symbol in front of the terms that remain inside.

For our calculator check...

answer with calculator showing that √(12*x^2*y^4)=2*√3*x*y^2

The calculator presents the answer a little bit different. However, I hope you can see that by doing some rearrangement to the terms that it matches with our final answer. Fantastic!

 


Example 8: Simplify the radical expression √(54*a^10*b^16*c^7).

For this problem, we are going to solve it in two ways. The goal is to show that there is an easier way to approach it especially when the exponents of the variables are getting larger.

Pairing Method: This is the usual way where we group the variables into two and then apply the square root operation to take the variable outside the radical symbol. You will see that for bigger powers, this method can be tedious and time consuming.

√(54*a^10*b^16*c^7)=3a^5b^8c^3√(6c)

 

"Division of Even Powers" Method: You can't find this name in any algebra textbook because I made it up. However, the key concept is there. The main approach is to express each variable as product of terms with even and odd exponents. If the term has an even power already, then you have nothing to do. Otherwise, you need to express it as some even power plus 1. Remember that getting the square root of "something" is equivalent to raising that "something" to a fractional exponent of 1/2. Simply put, divide the exponent of that "something" by 2.

check markThat's the reason why we want to express them with even powers since any even number is divisible by 2.

 

note that  c^7=c^(6+1)=c^6*c^1

That was better!


Example 9: Simplify the radical expression √(400h^4k^9m^7n^13).

Let's simplify this expression by first rewriting the odd exponents as powers of an even number plus 1.

For the number in the radicand, I see that 400 = 202.

final answer for √(400h^4k^9m^7n^13) = 20hk^4m^3n^6√(hkmn)

 


Example 10: Simplify the radical expression √(147w^6q^7q^27)
.

Express the odd powers as even numbers plus 1 then apply the square root to simplify further.

√(147w^6q^7q^27)=7w^3q^3r^13√(3qr)
=

 


Example 11: Simplify the radical expression square root of 32.

Try this on paper. Compare your answer to the solution below by clicking the mouse icon.

detailed solution to example example 11: sqrt(32)

 


Example 12: Simplify the radical expression square root of 125.

Try this on paper. Compare your answer to the solution below by clicking the mouse icon.

detailed solution to square root of 125

 

 


Example 13: Simplify the radical expression square root of (80x^3yz^5).

Try this on paper. Compare your answer to the solution below by clicking the mouse icon.

solution to the radical expression square root of (80x^3yz^5)

 

 


Example 14: Simplify the radical expression square root of (18m^11n^12k^13).

Try this on paper. Compare your answer to the solution below by clicking the mouse icon.

solution to the radical square root of (18m^11n^12k^13)

 


 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

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