Find practice problems and solutions for these concepts at Simplifying Radicals Practice Problems.
This lesson defines radicals and shows you how to simplify them. You will also learn how to add, subtract, multiply, and divide radicals.
What Is a Radical?
You have seen how the addition in x + 5 = 11 can be undone by subtracting 5 from both sides of the equation. You have also seen how the multiplication in 3x = 21 can be undone by dividing both sides by 3.Taking the square root (also called a radical) is the way to undo the exponent from an equation like x^{2 }= 25.
The exponent in 7^{2} tells you to square 7. You multiply 7 · 7 and get 7^{2}= 49.
The radical sign √ in √36 tells you to find the positive number whose square is 36. In other words, √36 asks: What number times itself is 36? The answer is √36 = 6 because 6 · 6 = 36.
The number inside the radical sign is called the radicand. For example, in √9, the radicand is 9.
Square Roots of Perfect Squares
The easiest radicands to deal with are perfect squares. Because they appear so often, it is useful to learn to recognize the first few perfect squares: 0^{2}= 0, 1^{2}= 1, 2^{2}= 4, 3^{2}= 9, 4^{2}= 16, 5^{2}= 25, 6^{2}= 36, 7^{2}= 49, 8^{2}= 64, 9^{2}= 81, 10^{2}= 100, 11^{2}= 121, and 12^{2}= 144. It is even easier to recognize when a variable is a perfect square, because the exponent is even. For example: x^{14}= x^{7}· x^{7},or (x^{7})^{2}, and a^{8}= a^{4}· a^{4}, or (a^{4})^{2}.
Example: √64 x2 y10
Write as a square. √8xy5 · 8xy5
Evaluate. 8xy^{5}
You could also have split the radical into parts and evaluated them separately:
Example: √64 x2 y10 

Split into perfect squares. 
√64 · x2 · y10 
Write as squares. 
√8 · 8 · √x · x · √y5 · y5 
Evaluate. 
8 · x · y^{5} 
Multiply together. 
8xy^{5} 
If your radical has a coefficient like 3√25, evaluate the square root before multiplying: 3√25 = 3 · 5 = 15.
Simplifying Radicals
Not all radicands are perfect squares. There is no whole number that, multiplied by itself, equals 5.With a calculator, you can get a decimal that squares to very close to 5, but it won't come out exactly. The only precise way to represent the square root of 5 is to write √5. It cannot be simplified any further.
There are three rules for knowing when a radical cannot be simplified any further:
 The radicand contains no factor, other than 1, that is a perfect square.
 The radicand is not a fraction.
 The radical is not in the denominator of a fraction.
When the Radicand Contains a Factor That Is a Perfect Square
To determine if a radicand contains any factors that are perfect squares, factor the radicand completely. All the factors must be prime. A number is prime if its only factors are 1 and the number itself. A prime number cannot be factored any further.
For example, here's how you simplify √12. The number 12 can be factored into 2 · 6. This is not completely factored because 6 is not prime. The number 6 can be further factored 2 · 3. The number 12 completely factored is 2 · 2 · 3.
The radical √12 can be written as √2 · 2 · 3. This can be split up into √2 · 2 · √3. Since √2 · 2 = 2, the simplified form of √12 is 2√3.
Example: √18 
Factor completely. 
√2 · 3 · 3 
Separate out the perfect square 3 · 3. 
√3 · 3 · √2 
Simplify. 
3√2 
Example: √60
Factor completely.
√6 · 10
Neither 6 nor 10 is prime. Both can be factored further.
√2 · 3 · 2 · 5
Separate out the perfect square
2 · 2. √2 · 2 · √3 · 5
Because √3 · 5 contains no perfect squares, it cannot be simplified further.
2√15
Example: √32 
Factor completely. 
√2 · 16 
The number 16 is not prime. It can be factored. 
√2 · 2 · 8 
The number 8 is not prime. It can be factored. 
√2 · 2 · 2 · 4 
The number 4 is not prime. It can be factored. 
√2 · 2 · 2 · 2 · 2 
You have two sets of perfect squares, 2 · 2 and 2 · 2. The square root of each is 2, so you have two square roots of 2. The square roots go outside the radical. You then multiply the numbers that are outside the radical.

2 · 2√2 
Simplify. The product of 2 times 2 gives you 4. 
4√2 
Shortcut: You may have noticed in the first step, √2 · 16, that 16 is a perfect square, and the square root of 16 is 4. This would have given you the answer 4√2. Use the shorter method whenever you see one.
Example: √50x3
Factor completely.
√2 · 5 · 5 · x · x · x
Separate the perfect square 5 · 5 and x · x.
√5 · 5 · √x · x · √2 · x
Simplify.
5x √2x
Example: √9x2y3
Rewrite the radicand as the product of perfect squares.
√9 x2 · y2 · y
Take out the square roots.
3xy √y
When the Radicand Contains a Fraction
The radicand cannot be a fraction. If you get rid of the denominator in the radicand, then you no longer have a fraction. This process is called rationalizing the denominator. Your strategy will be to make the denominator a perfect square. To do that, you multiply the denominator by itself.However, if you multiply the denominator of a fraction by a number, you must multiply the numerator of the fraction by the same number. Take a look at the following examples.
Example:
Make the denominator a perfect square.
Take out the square roots. One is a perfect square and so is 2 · 2.
Example:
Make the denominator a perfect square.
The number 1 is considered a factor of all numbers. If the numerator does not contain a perfect square, then 1 will be the perfect square and will be in the numerator. Take the square root of 1 in the numerator and 3 · 3 in the denominator. The product of 2 · 3 will give you 6 for the radicand.
Example:
Make the denominator a perfect square.
Take the square roots.
When a Radical Is in the Denominator
When you have a radical in the denominator, the expression is not in simplest form. The expression contains a radical in the denominator. To get rid of the radical in the denominator, rationalize the denominator. In other words, make the denominator a perfect square. To do that, you need to multiply the denominator by itself.
Example: 
Simplify. 

The number 9 is a perfect square. 

Example: 
Rationalize the denominator. 

Simplify. 

Take the square root of 4. 

Example: 
Rationalize the denominator. 

Simplify. 

You aren't finished yet because both radicands contain perfect squares. 


Take the square root of 4. 

Finished? Not quite. You can divide 2 into 2, or cancel the 2's. 
√3 
Adding and Subtracting Radicals
You can add and subtract radicals if the radicands are the same. For example, you can add 3√2 and 5√2 because the radicands are the same. To add or subtract radicals, you add the number in front of the radicals and leave the radicand the same.When you add 3√2 + 5√2, you add the 3 and the 5, but the radicand √2 stays the same. The answer is 8√2.
Tip
You can add or subtract radicals only when the radicand is the same. You add radicals by adding the number in front of the radicals and keeping the radicand the same. When you subtract radicals, you subtract the numbers in front of the radicals and keep the radicand the same.

Example: 2√5 + 7√5 
Add the numbers in front of the radicals. 
9√5 
Example: 11√5 – 4√5 
Subtract the numbers in front of the radicals. 
7√5 
Example: 4√3 + 2√5 + 6√3 
You can add only the radicals that are the same. 
10√3 + 2√5 
Example: 5√8 + 6√8 
Add the radicals. 
11√8 
But √8 contains a factor that is a perfect square, so you aren't finished because your answer is not in simplest form. 
11√2 · 4 
Take out the square root of 4. 
2 · 11√2 
Simplify. 
22√2 
Multiplying and Dividing Radicals
To multiply radicals like 4√3 by 2√2, you multiply the numbers in front of the radicals: 4 times 2. Then multiply the radicands: 3 times 2. The answer is 8√6.
Example: 5√3 · 2√2
Multiply the numbers in front of the radicals. Then multiply the radicands.
10√6
Example: 2√6 · 3√3
Multiply the numbers in front of the radicals. Then multiply the radicands.
6√18
However, you are not finished yet because 18 contains the factor 9, which is a perfect square.
6√2 · 9
Take out the square root of 9.
3 · 6√2
Simplify.
18√2
Tip
When you multiply or divide radicals, the radicands do not have to be the same.

To divide the radical 4√6 by 2√3, divide the numbers in front of the radicals. Then divide the radicands.The answer is 2√2.
Example:
Divide the numbers in front of the radical. Then divide the radicands.
2√3
Example:
Divide the numbers in front of the radicals. Then divide the radicands.
2√4
However, you aren't finished yet because 4 is a perfect square.
Take the square root of 4.
2 · 2
Simplify. 4
Tip
If there is no number in front of the radical sign, it is assumed to be 1.

Skill Building until Next Time
Think about the formula for the area of a rectangle: A = lw. Why do you use square when you multiply a number by itself? What word would you use when multiplying a number by itself three times? Why isn't there a commonly used word for multiplying a number by itself four times?

Find practice problems and solutions for these concepts at Simplifying Radicals Practice Problems.
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