Question

Find the square root of the following equation and choose the correct option from the given choices. $49\left( {{x}^{2}}-2xy+{{y}^{2}} \right)$
(a) $7\left| x-y \right|$
(b) $7\left( x+y \right)\left( x-y \right)$
(c) $7{{\left( x+y \right)}^{2}}$
(d) $7{{\left( x-y \right)}^{2}}$

Answer 1

Hint:We will use the formula ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$ to solve this question. We will simply and find the square root of $49\left( {{x}^{2}}-2xy+{{y}^{2}} \right)$ by using basic mathematical operations.

Complete step-by-step answer:
It is given in the question to find the root of the expression $49\left( {{x}^{2}}-2xy+{{y}^{2}} \right)$. We know that ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$,
where $a=x,b=y$,
so we can write $\left( {{x}^{2}}-2xy+{{y}^{2}} \right)$ as ${{\left( x-y \right)}^{2}}$, and thus we get
$49{{\left( {{\left( x-y \right)}^{2}} \right)}^{2}}$
We know that ${{(x^a)}}^b$=${(x^{ab})}$ applying this property we get,
$49{{\left( x-y \right)}^{4}}$
Now, we will find the square root of $49{{\left( x-y \right)}^{4}}$ by using basic mathematical operations. That is,
$\sqrt{49{{\left( x-y \right)}^{4}}}$
Simplifying further, we get $\sqrt{{{\left( 7{{\left( x-y \right)}^{2}} \right)}^{2}}}$
Considering square root as power of $\dfrac{1}{2}$, we get ${{\left( 7{{\left( x-y \right)}^{2}} \right)}^{2\times \dfrac{1}{2}}}$
Cancelling one of the square power, we finally get $7{{\left( x-y \right)}^{2}}$
Therefore, the root of the given expression $49\left( {{x}^{2}}-2xy+{{y}^{2}} \right)$ is $7{{\left( x-y \right)}^{2}}$, therefore the correct answer is option d).

Note: Student may start solving this question wrong way by simply finding the square of $\left( {{x}^{2}}-2xy+{{y}^{2}} \right)$ and then multiply it by 49 to have the final terms of which we have to find a square root. But this process is highly not recommended because it can be solved in just a few lines and by using basic mathematical operations without making any error by using the above process. Also, student is recommended to memorize all the basic mathematical identities and formulas, like the one used above ${{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab$, which makes it easy to solve some unnecessary steps. These identities are general identities, which need to be memorized carefully taking care about the + and the – signs.