How to divide surds

The use of surds goes as far back as 800 CE. Originally coined by a Persian mathematician by the name of al-Khwarizmi, he first called rational and irrational numbers ‘audible’ and ‘inaudible’, respectively.

Since then, the term surds has been used to describe the simplest forms of any irrational number, and they can be used in all basic mathematical operations such as addition, subtraction, multiplication, and division. 

However, it’s often the division of surds that trips people up. Not only can surd divisions look visually intimidating, but it can also take some time to practice the entire process. But once you understand the basic laws and rules of surds, we can guarantee that it will be far simpler than you initially thought.

But before we dive into the details of that process, let’s first review exactly what a surd is.

Surds – often called a radical – are irrational numbers that cannot be simplified to an integer and are, thus, expressed in a square root of a number. For instance, √2 is a surd.

By definition, irrational numbers are numbers that cannot be expressed as a fraction or ratio of two numbers and, when written in decimal form, contain digits that do not recur and thus go on forever. The most common example of an irrational number is pi(π).

Therefore, surds allow us to express an irrational number shortly and concisely, which cannot be done in decimal form. As such, they have important applications in mathematics where irrational numbers are concerned. 

Dividing surds by surds is incredibly straightforward and can be summarized using the following formula: √a ÷ √b = √(a ÷ b). Therefore, to solve a division equation, follow the subsequent steps:

• Step 1: Consolidate the square roots into one surd using the formula: √a ÷ √b = √(a ÷ b)
• Step 2: Solve the division problem inside the square root
• Step 3: Simplify the answer to its simplest form 

As you can see, it’s not complicated and is exactly the same as when dividing whole numbers. Let’s take a look at a few examples.

Example 1

What is √50 ÷ √5? 

• Step 1: Consolidate the square roots into one surd using the formula √a ÷ √b = √(a ÷ b)

This means that our surd equation can be expressed as √(50 ÷ 5).

• Step 2: Solve the division problem inside the square root

50 divided by 5 is 10. Therefore, we can rewrite the surd as √10.

• Step 3: Simplify the answer to its simplest form

The surd √10 cannot be simplified any further, and you have thus reached your final answer.

This means that √50 ÷ √5 =√10.

Example 2

What is the answer to √24 ÷ √3?

• Step 1: Consolidate the square roots into one surd using the formula √a ÷ √b = √(a ÷ b)

This means that the surd equation can be expressed as √(24 ÷ 3).

• Step 2: Solve the division problem inside the square root

24 divided by 3 is 8. Therefore, we can rewrite the surd as √8.

• Step 3: Simplify the answer to its simplest form

The surd √8 can be rewritten as √4 x √2, which can be simplified to 2 x √2. Expressed in its most simplest form, it becomes 2√2. This is your final answer.

Therefore, √24 ÷ √3 =2√2.

Example 3

Solve the equation √18 ÷ √6.

• Step 1: Consolidate the square roots into one surd using the formula √a ÷ √b = √(a ÷ b)

This means that the surd equation can be expressed as √(18 ÷ 6).

• Step 2: Solve the division problem inside the square root

18 divided by 6 is 3. Therefore, we can rewrite the surd as √3.

• Step 3: Simplify the answer to its simplest form

The surd √3 cannot be simplified any further, and you have therefore reached your final answer.

This means that √18 ÷ √6 =√3.

These examples show that the simple three-step process can be used for any division problem where you are dividing one surd by another. But what if you are dividing surds that have an integer before them? Let’s find out.

When dividing surds with whole numbers at the front, they will be written in the format a√b. You must use a different formula to solve a division problem with two surds that look like this. It is as follows: a√b​ ÷ c√d​ = (a ÷ c)√(b ÷ d). 

As such, the steps involved in solving a problem like this are also different:

• Step 1: Substitute the values of ‘a’, ‘b’, ‘c’, and ‘d’ into the formula a√b​ ÷ c√d​ = (a ÷ c)√(b ÷ d)
• Step 2: Divide the coefficients ‘a’ and ‘c’ together
• Step 3: Divide the terms inside the square root ‘b’ and ‘d’ together 
• Step 4: Combine the results of step 2 and step 3 together
• Step 5: Simplify the answer to its simplest form

Effectively, you are first dividing the numbers outside the square root sign together, then dividing the numbers inside the square root sign together, and finally combining the answer to those steps together to achieve your final solution. Let’s see this process in action.

Example 4

Solve 8√30 ÷ 4√​10.

• Step 1: Substitute the values of ‘a’, ‘b’, ‘c’, and ‘d’ into the formula a√b​ ÷ c√d​ = (a ÷ c)√(b ÷ d)

Substituting in the values would give you (8 ÷ 4)√(30 ÷ 10).

• Step 2: Divide the coefficients ‘a’ and ‘c’ together

Dividing the coefficients together gives 2 (= 8 ÷ 4).

• Step 3: Divide the terms inside the square root ‘b’ and ‘d’ together 

Dividing the terms inside the square root gives √3 (= 30 ÷ 10).

• Step 4: Combine the results of step 2 and step 3 together

Combining steps 2 and 3 together gives an answer of 2√3.

• Step 5: Simplify the answer to its simplest form

The solution cannot be simplified any further; thus, you have reached the final answer.

Therefore, 8√30 ÷ 4√​10 = 2√3.

Example 5

Solve 10√8 ÷ 2√​2.

• Step 1: Substitute the values of ‘a’, ‘b’, ‘c’, and ‘d’ into the formula a√b​ ÷ c√d​ = (a ÷ c)√(b ÷ d)

Substituting in the values would give you (10 ÷ 2)√(8 ÷ 2).

• Step 2: Divide the coefficients ‘a’ and ‘c’ together

Dividing the coefficients together gives 5 (= 10 ÷ 2).

• Step 3: Divide the terms inside the square root ‘b’ and ‘d’ together 

Dividing the terms inside the square root gives √4 (= 8 ÷ 2).

• Step 4: Combine the results of step 2 and step 3 together

Combining steps 2 and 3 together gives an answer of 5√4.

• Step 5: Simplify the answer to its simplest form

The surd √4 can be simplified to equal 2. Therefore, substituting 2 back into the answer gives you 5 x 2 = 10. This is your final answer.

As a result, 10√8 ÷ 2√​2 = 10.

Example 6

Solve 48√6 ÷ 8√​3.

• Step 1: Substitute the values of ‘a’, ‘b’, ‘c’, and ‘d’ into the formula a√b​ ÷ c√d​ = (a ÷ c)√(b ÷ d)

Substituting in the values would give you (48 ÷ 8)√(6 ÷ 3).

• Step 2: Divide the coefficients ‘a’ and ‘c’ together

Dividing the coefficients together gives 6 (= 48 ÷ 8).

• Step 3: Divide the terms inside the square root ‘b’ and ‘d’ together 

Dividing the terms inside the square root gives √2 (= 6 ÷ 3).

• Step 4: Add the results of steps 2 and step 3 together

Combining steps 2 and 3 together gives an answer of 6√2.

• Step 5: Simplify the answer to its simplest form

In this case, you cannot simplify the solution any further and have therefore reached the final answer.

This means that 48√6 ÷ 8√​3 = 6√2.

To divide an integer by a surd, follow the steps outlined below:

• Step 1: Express the division equation as a fraction
• Step 2: Determine the rationalizing factor 
• Step 3: Multiply the numerator and denominator by the rationalizing factor
• Step 4: Simplify the answer to its simplest form

Before we get into practice problems that show this in action, let’s first explain the steps so you understand the process completely. 

• Step 1: Express the division equation as a fraction

All division problems can be expressed as a fraction, with the dividend being the numerator and the divisor being the denominator. The same applies to when dividing surds. When dividing an integer by a surd, this becomes a necessary step.

• Step 2: Determine the rationalizing factor

You cannot leave a surd in the denominator of a fraction as it is an irrational number. Therefore, you must rationalize the denominator, which means getting rid of it. To do so, you must determine what the rationalizing factor is. In any case, it is simply the denominator.

• Step 3: Multiply the numerator and denominator by the rationalizing factor

This is self-explanatory. Multiplying the numerator and denominator by the rationalizing factor will remove the surd from the denominator, which is the goal. It should be noted that multiplying surds follows the same process as when multiplying integers.

• Step 4: Simplify the answer to its simplest form

Finally, you must check whether the fraction can be simplified any further.

Example 1

Solve 8 ÷ √3.

• Step 1: Express the division equation as a fraction

Converting the equation to a fraction gives the following result: 8/√3

• Step 2: Determine the rationalizing factor

Since the denominator is √3, this means that the rationalizing factor is also √3.

• Step 3: Multiply the numerator and denominator by the rationalizing factor

Numerator: 8 x √3 = 8√3

Denominator: √3 x √3 = (√3)²

Resulting fraction: 8√3/(√3)²

• Step 4: Simplify the answer to its simplest form

The numerator cannot be simplified any further, but the denominator can. When you multiply surds of the same value together, the square roots cancel each other out, leaving an integer behind. 

Therefore, simplifying the denominator (√3)² equals 3. As such, the final answer is as follows:

8 ÷ √3 = 8√3/3.

Example 2

Rationalize the denominator of the following surd: 1/√5

• Step 1: Express the division equation as a fraction

This problem is already stated as a fraction, so you can skip this step.

• Step 2: Determine the rationalizing factor

Since the denominator is √5, this means that the rationalizing factor is also √5.

• Step 3: Multiply the numerator and denominator by the rationalizing factor

Numerator: 1 x √5 = 1√5

Denominator: √5 x √5 = (√5)²

Resulting fraction: 1√5/(√5)²

• Step 4: Simplify the answer to its simplest form

The numerator can be simplified from 1√5 to √5. The denominator can be simplified from (√5)² to 5. As such, the final answer is as follows:

√5/5.

Example 3

Solve 6 ÷ √13.

• Step 1: Express the division equation as a fraction

Converting the equation to a fraction gives you 6/√13.

• Step 2: Determine the rationalizing factor

Since the denominator is √13, this means that the rationalizing factor is also √13.

• Step 3: Multiply the numerator and denominator by the rationalizing factor

Numerator: 6 x √13 = 6√13

Denominator: √13 x √13 = (√13)²

Resulting fraction: 8√13/(√13)²

• Step 4: Simplify the answer to its simplest form

The numerator cannot be simplified any further. The denominator simplifies from (√13)² to 13. Therefore, the final answer is as follows:

6 ÷ √13 = 6√13/13. 

Surds are the simplest forms in which to express an irrational number. Instead of writing a long decimal form number that never ends, surds provide a short and concise alternative that can still give the necessary accuracy for calculations. 

Dividing surds depends on whether you’re dividing a surd by another surd or an integer by a surd. However, if you follow the steps outlined in this article, you’ll be well-equipped to tackle any problem with ease.