Question

# Find the rationalizing factor: $\sqrt{18}$ ; $5\sqrt{3}$

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Hint: In order to solve this problem, we need to understand the term rationalizing factor. The numbers which can be expressed in the form of a fraction of two integers are called rational numbers. And the numbers which cannot be expressed in the form of a fraction are called an irrational number.

Let's first understand the rationalizing factor.
The numbers which can be expressed in the form of a fraction of two integers are called rational numbers. And the numbers which cannot be expressed in the form of a fraction are called an irrational number.
The smallest fraction that we need to multiply in order to get a rational number is called the rationalizing factor.
Let’s consider the first example.
$\sqrt{18}$
Now we can see that this is an irrational word.
But we can factorize the term inside the square root.
By factorizing we get,
$\sqrt{18}=\sqrt{2\times 3\times 3}$
Solving it further we get,
$\sqrt{18}=3\sqrt{2}$ ,
Now we cannot factorize it further, so in order to make the number rational, we need to remove the square root.
To remove the square root, we need to multiply the number by the same square root.
By multiplying by $\sqrt{2}$ , we get,
$\sqrt{18}\times \sqrt{2}=3\sqrt{2}\times \sqrt{2}=6$
Now 6 is a rational number.
Hence the rationalizing factor is $\sqrt{2}$ .
Let’s consider the second example.
$5\sqrt{3}$
Now we can see that this is an irrational word.
But we cannot factorize the term inside the square root.
So, in order to make the number rational we need to remove the square root.
To remove the square root, we need to multiply the number by the same square root.
By multiplying by $\sqrt{3}$ , we get,
$5\sqrt{3}\times \sqrt{3}=5\sqrt{3}\times \sqrt{3}=15$
Now 15 is a rational number.
Hence the rationalizing factor is $\sqrt{3}$.

Note: Rationalizing factor is always the smallest factor that needs to be multiplied. So, we always need to factorize so as to get the rationalizing factor as small as possible. Also, not every number in the square root is irrational. The perfect squares in the square root are rational numbers.