Question

How do you simplify \[\sqrt{2}\left( \sqrt{2}-\sqrt{3} \right)\]?

Answer 1

Hint: In this problem, we have to simplify the given expression. We can first multiply the term outside the bracket to the terms inside the bracket. We know that the product of square roots is equal to the square root of the product of the radicands, where the radicands are non-negative numbers. We can apply this formula in the expression to get a simplified form of the given expression.

Complete step by step answer:
We know that the given expression to be simplified is,
\[\sqrt{2}\left( \sqrt{2}-\sqrt{3} \right)\]
We can multiply the term outside the bracket to the terms inside the bracket, we get
\[\Rightarrow \sqrt{2}\times \sqrt{2}-\sqrt{2}\times \sqrt{3}\] …. (1)
We know that the product of square roots is equal to the square root of the product of the radicands, where the radicands are non-negative numbers.
\[\sqrt{a}\times \sqrt{b}=\sqrt{a\times b}\]
Now we can apply the above formula n expression (1), we get
\[\Rightarrow \sqrt{2}\times \sqrt{2}-\sqrt{2}\times \sqrt{3}=\sqrt{2\times 2}-\sqrt{2\times 3}\]
Now we can multiply the terms inside the root in the right-hand side, we get
\[\begin{align}
  & \Rightarrow \sqrt{4}-\sqrt{6} \\
 & \Rightarrow 2-\sqrt{6}\text{ }\because \sqrt{4}=2 \\
\end{align}\]
Therefore, the simplified form of \[\sqrt{2}\left( \sqrt{2}-\sqrt{3} \right)\] is \[2-\sqrt{6}\].

Note:
Students make mistakes while adding the terms or multiplying the terms inside or outside the root, which should be concentrated. We know that the product of square roots is equal to the square root of the product of the radicands, where the radicands are non-negative numbers. We can also write the exact root values if we are asked to find the exact value for the given expression.