Answer

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**Hint:**Finding the value of \[{(m + n - p)^2}\], we start by adding $\sqrt p $ to \[\sqrt m + \sqrt n - \sqrt p = 0\] on both sides. Then, we square the obtained result on both sides and also use ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$. At last simplify the results to get the value of \[{(m + n - p)^2}\].

**Complete step by step answer:**

Given that the value of \[\sqrt m + \sqrt n - \sqrt p = 0\]. And we need to find the value of \[{(m + n - p)^2}\]

Let us consider \[\sqrt m + \sqrt n - \sqrt p = 0\]

Now, we add both sides on the above equation with $\sqrt p $

$ \Rightarrow $ \[\sqrt m + \sqrt n - \sqrt p + \sqrt p = \sqrt p \]

$ \Rightarrow $ \[\sqrt m + \sqrt n = \sqrt p \]

Now, we square on both sides of the above equations to get as follows.

$ \Rightarrow $ \[{\left( {\sqrt m + \sqrt n } \right)^2} = {\left( {\sqrt p } \right)^2} \cdots \cdots \cdots \left( 1 \right)\]

We know that ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$. We use these results in equation (1) comparing the formula we get, $a = \sqrt m $ ,$b = \sqrt n $ put these values in the formula. Now we have

$ \Rightarrow $ $m + n + 2\sqrt {mn} = p$

Now we add both sides with $ - p - 2\sqrt {mn} $ on the above equation then we get,

$ \Rightarrow $ $m + n + 2\sqrt {mn} - p + 2\sqrt {mn} = p - p + 2\sqrt {mn} $

$ \Rightarrow $ \[m + n - p = - 2\sqrt {mn} \cdots \cdots \cdots \left( 2 \right)\]

Now, we square on both sides of the above equation to get as follows

$ \Rightarrow $ \[{\left( {m + n - p} \right)^2} = {\left( { - 2\sqrt {mn} } \right)^2}\]

Separating the power using the exponent law of power that is ${\left( {ab} \right)^m} = {a^m} \times {b^m}$ then comparing the ${\left( { - 2\sqrt {mn} } \right)^2}$ with the law we have $a = - 2$ and $b = \sqrt {mn} $ putting these value in the exponent law we get,

$ \Rightarrow {\left( {m + n - p} \right)^2} = {\left( { - 2} \right)^2} \times {\left( {\sqrt {mn} } \right)^2}$

We know that the square of any negative number is positive. Therefore square of $ - 2$ will be four and square of $\sqrt {mn} $ will be $mn$ and we get,

$ \Rightarrow $ \[{\left( {m + n - p} \right)^2} = 4mn\]

**Hence the value of \[{\left( {m + n - p} \right)^2}\] is $4mn$.**

**Note:**

Simplifying the equation, you have to keep in mind the expression whose value you have to find and also the calculation like squaring, adding, and subtracting to get the final result. Moreover, you need to use the equation whatever given to you in the question.

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