Question

If the value of $ \sqrt{m}+\sqrt{n}-\sqrt{p}=0 $ . Find the value of $ {{\left( m+n-p \right)}^{2}} $ ?

Answer 1

Hint: We start by adding $ \sqrt{p} $ to $ \sqrt{m}+\sqrt{n}-\sqrt{p}=0 $ on both sides. We square the obtained result on both sides. We use the results $ {{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab $ and $ {{\left( \sqrt{x} \right)}^{2}}=x $ while squaring on both sides. After squaring we add and subtract terms on both sides to get m+n-p on one side. Once we get
m+n-p on one side, we square on both sides and use $ {{\left( \sqrt{x} \right)}^{2}}=x $ to get the value of $ {{\left( m+n-p \right)}^{2}} $ .

Complete step-by-step answer:
We have given that the value of $ \sqrt{m}+\sqrt{n}-\sqrt{p}=0 $ . We need to find the value of $ {{\left( m+n-p \right)}^{2}} $ .
Let us consider $ \sqrt{m}+\sqrt{n}-\sqrt{p}=0 $ .
Now, we add both sides with $ \sqrt{p} $ .
 $ \sqrt{m}+\sqrt{n}-\sqrt{p}+\sqrt{p}=0+\sqrt{p} $ .
 $ \sqrt{m}+\sqrt{n}=\sqrt{p} $ .
Now, we square on both sides to get as follows.
 $ {{\left( \sqrt{m}+\sqrt{n} \right)}^{2}}={{\left( \sqrt{p} \right)}^{2}} $ ---(1).
We know that $ {{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab $ and $ {{\left( \sqrt{x} \right)}^{2}}=x $ . We use these results in equation (1).
 $ {{\left( \sqrt{m} \right)}^{2}}+{{\left( \sqrt{n} \right)}^{2}}+\left( 2.\sqrt{m}.\sqrt{n} \right)=p $ .
 $ m+n+2\sqrt{mn}=p $ .
Now we add both sides with $ \left( -p-2\sqrt{mn} \right) $ .
 $ m+n+2\sqrt{mn}+\left( -p-2\sqrt{mn} \right)=p+\left( -p-2\sqrt{mn} \right) $ ---(2).
We know that $ +\times -=- $ . We use this result in (2).
 $ m+n+2\sqrt{mn}-p-2\sqrt{mn}=p-p-2\sqrt{mn} $ .
 $ m+n-p=-2\sqrt{mn} $ .
Now, we square on both sides to get as follows.
\[{{\left( m+n-p \right)}^{2}}={{\left( -2\sqrt{mn} \right)}^{2}}\].
We know that $ {{\left( -ab \right)}^{2}}={{a}^{2}}.{{b}^{2}} $ .
 $ {{\left( m+n-p \right)}^{2}}={{2}^{2}}.{{\left( \sqrt{mn} \right)}^{2}} $ .
We know that $ {{\left( \sqrt{x} \right)}^{2}}=x $ .
 $ {{\left( m+n-p \right)}^{2}}=4.mn $
 $ {{\left( m+n-p \right)}^{2}}=4mn $ .
∴ The value of $ {{\left( m+n-p \right)}^{2}} $ is 4mn.

Note: We need to make sure that the sign related calculations like $ \left( - \right)\times \left( + \right)=\left( - \right) $ , $ {{\left( - \right)}^{2}}=+ $ are to be done without any mistakes. We need to do squaring or cubing with the requirements of the problem. We opted to do squaring to begin the process as we can see there are square roots present for variables m, n, and p at the start of the problem and we don’t have any square roots for m, n and p at the final requirement of the problem. Similarly, we can be asked to find the value of $ {{\left( m+p-n \right)}^{2}} $ , $ {{\left( n+p-m \right)}^{2}} $ .