Explainer: Simplifying Radicals

In this explainer, we will learn how to perform mathematical operations on radicals and how to utilize the conjugate to rationalize the denominator of a radical and simplify radical expressions.

Before we start looking at how to simplify radical expressions, let us first clearly define what we mean when we talk about radicals and define some associated mathematical language.

Firstly, the word radical describes the β€œroot sign”; the number contained within the root sign is called the radicand, and the little number on the outside of the radical is call the index which tells us if we are dealing with a square root, a cube root, or otherwise. These terms have been labeled in the following picture.

When we want to simplify radicals, we want to minimize the size of the radicand. In order to do this, we first need to introduce two key rules.

Key Rules: Products and Quotients of Radicals

  1. βˆšπ‘ŽΓ—βˆšπ‘=βˆšπ‘Žπ‘
  2. βˆšπ‘Žβˆšπ‘=ο„žπ‘Žπ‘

It is worth noting that, generally, when simplifying radicals, we tend to focus on square roots, though the methods can be applied to radicals of any index (provided simplification calculations only contain radicals with the same index).

In order to minimize the radicand, we need to use the reverse of rule one. Let us consider the radical expression √32.

The number 32 is a multiple of 16 which is a perfect square, so, we can rewrite √32 as √16Γ—2. If we then apply rule one in reverse, we can see that √32=√16Γ—βˆš2, and, as 16 is a perfect square, we can simplify this to find that √32=4√2.

You may notice that 32 is also a multiple of 4 which is another perfect square. We could then rewrite √32 as √4Γ—8 which is the same as √4Γ—βˆš8, which simplifies to 2√8. Here, however, we have not minimized the radicand as 8 is still a multiple of a perfect square. So, we can continue the process: 2Γ—βˆš8=2Γ—βˆš4Γ—βˆš2.

This simplifies to 2Γ—2Γ—βˆš2=4√2, as before.

It is generally easiest to identify the largest perfect square that is a factor of the number you are simplifying and then the method will be more efficient. In case, however, you happen to start with a perfect square that is not the largest factor, it is always worth checking your resulting expression to see if it has any more perfect squares as factors (in other words, can the radicand be reduced further?).

Let us now look at a more formal example.

Example 1: Simplifying Radicals

Express √640 in the form π‘Žβˆšπ‘ where π‘Ž and 𝑏 are integers and 𝑏 takes the least possible value.

Answer

With these questions, it is often helpful to write out a list of perfect squares: 1,4,9,16,25,36,49,64,81,100,121,144,….

From this list, we can see that the largest perfect square that is a factor of 640 is 64. We can then rewrite the radical as follows: √640=√64Γ—10=√64Γ—βˆš10=8√10.

Now, let us look at a second example where the radical is given in the from π‘Žβˆšπ‘, but 𝑏 is a multiple of a perfect square.

Example 2: Simplifying Radicals

Express 3√45 in the form π‘Žβˆšπ‘ where π‘Ž and 𝑏 are integers and 𝑏 takes the least possible value.

Answer

Here, we need to rewrite the expression by simplifying the radical √45 and then multiplying the result by 3: 3√45=3Γ—ο€»βˆš9Γ—5=3Γ—ο€»βˆš9Γ—βˆš5=3Γ—ο€»3Γ—βˆš5=9√5.

Finally, let us look at couple of examples where the simplification of the radical is only part of the question.

Example 3: Simplifying Expressions Containing Multiple Radicals

Express √3+√12βˆ’βˆš75 in its simplest form.

Answer

First, we need to look at each of the radicals separately and see if they can be simplified. We can see that √3 can not be reduced further as 3 is prime. The √12, however, can be simplified as 4 is a factor of 12. So, √12=√4Γ—3=√4Γ—βˆš3=2√3.

Now, if we look at √75 we can see that this too simplifies further as it is a multiple of 25. So, we find that √75=√25Γ—3=√25Γ—βˆš3=5√3.

So, the original expression is the same as √3+2√3βˆ’5√3.

As we have three terms containing radicals of equal index, and equal radicand, we can simplify this expression to get βˆ’2√3.

Example 4: Simplifying Expressions Containing Multiple Radicals

Express √75+√28βˆ’βˆš27 in its simplest form.

Answer

To solve this question, we need to consider each of the radicals individually and see if they can be simplified. First, we have √75 which can be simplified as 28 is a multiple of 25: √75=√25Γ—3=√25Γ—βˆš3=5√3.

Secondly, we have √28 which can be simplified as: √28=√4Γ—7=√4Γ—βˆš7=2√7.

Finally, √27 can be simplified as 27 is a multiple of 9: √27=√9Γ—3=√9Γ—βˆš3=3√3.

We can, therefore, rewrite the expression as 5√3+2√7βˆ’3√3, and we can combine the first and last terms, as they contain the same radicand, to get 2√3+2√7.

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