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The square root of $49{{\left( {{x}^{2}}-2xy+{{y}^{2}} \right)}^{2}}$ is:
(a) $7\left| x-y \right|$
(b) $7\left( x+y \right)\left( x-y \right)$
(c) $7{{\left( x+y \right)}^{2}}$
(d) $7{{\left( x-y \right)}^{2}}$

Last updated date: 20th Jun 2024
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Hint: Try to simplify the expression that is given in the question. Make it as simplified as it can be and then apply square root function to this simplified expression.

In the question, we have to find the square root of the expression $49{{\left( {{x}^{2}}-2xy+{{y}^{2}} \right)}^{2}}$. If we try to do square root of this expression, we will be able to find it’s square root but we will not be able to match our answer with any of the options. So, there is a need to simplify the expression so that we can easily find the square root of the given expression and finally, can match our answer with the options given in this question.
Before proceeding with the simplifying process, we must know a formula,
${{\left( x-y \right)}^{2}}={{x}^{2}}-2xy+{{y}^{2}}$
Using this formula, we can also say that,
${{x}^{2}}-2xy+{{y}^{2}}={{\left( x-y \right)}^{2}}..........\left( 1 \right)$
If we substitute ${{x}^{2}}-2xy+{{y}^{2}}$ as ${{\left( x-y \right)}^{2}}$ in any expression, we will get a perfect square term i.e. ${{\left( x-y \right)}^{2}}$ in that particular expression. Since that expression will contain a perfect square term, it will be easy to find the square root of that term.
So, substituting ${{x}^{2}}-2xy+{{y}^{2}}={{\left( x-y \right)}^{2}}$ from equation $\left( 1 \right)$ in the expression given in the question i.e. $49{{\left( {{x}^{2}}-2xy+{{y}^{2}} \right)}^{2}}$, we get,
  & 49{{\left( {{x}^{2}}-2xy+{{y}^{2}} \right)}^{2}}=49{{\left( {{\left( x-y \right)}^{2}} \right)}^{2}} \\
 & \Rightarrow 49{{\left( {{x}^{2}}-2xy+{{y}^{2}} \right)}^{2}}={{\left( 7 \right)}^{2}}{{\left( {{\left( x-y \right)}^{2}} \right)}^{2}} \\
In the question, we are asked to find the square root of this expression. Applying square root function on this expression, we get,
  & \sqrt{{{\left( 7 \right)}^{2}}{{\left( {{\left( x-y \right)}^{2}} \right)}^{2}}} \\
 & \Rightarrow \pm 7{{\left( x-y \right)}^{2}} \\
We got two answers for the square root of $49{{\left( {{x}^{2}}-2xy+{{y}^{2}} \right)}^{2}}$. But since only one of the two answers is there in the options, we will mark only that option as our answer.
Hence, the answer is option (d).

Note: There is a possibility that the examiner may have given both $\pm 7{{\left( x-y \right)}^{2}}$ in the options. So, in that case, we have to mark both the options as our answer. Also, even without solving completely, one can find the answer from the options by eliminating the other options if one can identify by looking at the question that ${{x}^{2}}-2xy+{{y}^{2}}$ can be also written as ${{\left( x-y \right)}^{2}}$.