Question

How do you simplify \[\dfrac{\sqrt{22}}{\sqrt{11}-\sqrt{66}}\]?

Answer 1

Hint: In this problem, we have to simplify and find the value of the given expression. We can first take the numerator, where we can write the number 22 inside the root as \[2\times 11\], we can then separate the root into two terms with the multiplication of the root formula. We can also change the denominator in the similar way as the numerator and we can cancel the common terms in the numerator and the denominator to get a simplified form.

Complete step by step answer:
We know that the given expression to be simplified is,
\[\dfrac{\sqrt{22}}{\sqrt{11}-\sqrt{66}}\]
We can now write the above expression as,
\[\Rightarrow \dfrac{\sqrt{2\times 11}}{\sqrt{1\times 11}-\sqrt{6\times 11}}\]
We can now separate the root terms in both the numerator and the denominator, using the multiplication of roots formula.
We know that the multiplication of roots formula is,
\[\Rightarrow \sqrt{xy}=\sqrt{x}\times \sqrt{y}\]
We can now apply this formula in the above step for both numerator and the denominator, we get
\[\Rightarrow \dfrac{\sqrt{2}\times \sqrt{11}}{\sqrt{11}\times \sqrt{1}-\sqrt{6}\times \sqrt{11}}\]
Now we can take the common terms outside in the denominator, we get
\[\Rightarrow \dfrac{\sqrt{2}\times \sqrt{11}}{\sqrt{11}\left( 1-\sqrt{6} \right)}\]
We can now cancel the similar terms in the numerator and the denominator, we get
\[\Rightarrow \dfrac{\sqrt{2}}{\left( 1-\sqrt{6} \right)}\]
Therefore, the simplified form of the given expression \[\dfrac{\sqrt{22}}{\sqrt{11}-\sqrt{66}}\] is \[\dfrac{\sqrt{2}}{\left( 1-\sqrt{6} \right)}\].

Note:
We should remember that we can separate the terms in root by using multiplication of roots formula. We should know that the multiplication of terms inside the root is equal to multiplication of roots with its individual terms. We should also remember that the value of root of 1 is 1.