Question

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Answer 1

Question :–

▪︎ If ${ \bold{ \: \: \sqrt{3} = 1.732 \: \: }}$ , then find the value of ${ \bold{ \: \: \sqrt{ \dfrac{2 - \sqrt{3} }{2 + \sqrt{3} } } = ? \: \: }}$

ANSWER :–

$\\ \longrightarrow { \red{ \boxed{ \bold{ \: \: \sqrt{ \dfrac{2 - \sqrt{3} }{2 + \sqrt{3} } } = \: \: \pm \: (0.268) \: \: }}}}$

EXPLANATION :–

GIVEN :–

$\\ { \bold{ \: \implies \sqrt{3} = 1.732 \: \: }}$

TO FIND :–

$\\ { \bold{ \: \: \implies \sqrt{ \dfrac{2 - \sqrt{3} }{2 + \sqrt{3} } } = ? \: \: }}$

SOLUTION :–

$\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \: \: \: \sqrt{ \dfrac{2 - \sqrt{3} }{2 + \sqrt{3} } } \: \: }}$

• Now Rationalization –

$\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \: \: \: \sqrt{ \dfrac{2 - \sqrt{3} }{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } } \: \: }}$

$\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \: \: \: \sqrt{ \dfrac{(2 - \sqrt{3} )^{2} }{ {(2)}^{2} - ( \sqrt{3})^{2} } } \: \: }}$

$\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \: \: \: \sqrt{ \dfrac{(2 - \sqrt{3} )^{2} }{4 - 3}} \: \: }}$

$\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \: \: \: \sqrt{ {(2 - \sqrt{3} )^{2} }} \: \: }}$

$\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \pm (2 - \sqrt{3}) }}$

• Now put the value of ${ \bold{ \: \sqrt{3} \: \: - }}$

$\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \pm \: (2 - 1.732) }}$

$\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \pm \: (0.268) }}$

Hence , ${ \bold{ \: \: \sqrt{ \dfrac{2 - \sqrt{3} }{2 + \sqrt{3} } } = \: \: \pm \: (0.268) \: \: }}$

$\\ \bigstar \: \: \large { \red{ \bold{ \underline{used \: \: formula} } : - }}$

$\\ { \blue{ \bold{(1) \: \: \: (a - b)(a + b) = {a}^{2} - {b}^{2} }}}$

$\\ { \blue{ \bold{(2) \: \: \: {(a -b)}^{2} = {a}^{2} + {b}^{2} - 2ab }}}$

Answer 2

Step-by-step explanation:

Question :–

▪︎ If { \bold{ \: \: \sqrt{3} = 1.732 \: \: }}

3

=1.732 , then find the value of \begin{lgathered}{ \bold{ \: \: \sqrt{ \dfrac{2 - \sqrt{3} }{2 + \sqrt{3} } } = ? \: \: }} \\\end{lgathered}

2+

3

2−

3

=?

ANSWER :–

\begin{lgathered}\\ \longrightarrow { \red{ \boxed{ \bold{ \: \: \sqrt{ \dfrac{2 - \sqrt{3} }{2 + \sqrt{3} } } = \: \: \pm \: (0.268) \: \: }}}} \\\end{lgathered}

⟶

2+

3

2−

3

=±(0.268)

EXPLANATION :–

GIVEN :–

\begin{lgathered}\\ { \bold{ \: \implies \sqrt{3} = 1.732 \: \: }} \\\end{lgathered}

⟹

3

=1.732

TO FIND :–

\begin{lgathered}\\ { \bold{ \: \: \implies \sqrt{ \dfrac{2 - \sqrt{3} }{2 + \sqrt{3} } } = ? \: \: }} \\\end{lgathered}

⟹

2+

3

2−

3

=?

SOLUTION :–

\begin{lgathered}\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \: \: \: \sqrt{ \dfrac{2 - \sqrt{3} }{2 + \sqrt{3} } } \: \: }} \\\end{lgathered}

=

2+

3

2−

3

• Now Rationalization –

\begin{lgathered}\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \: \: \: \sqrt{ \dfrac{2 - \sqrt{3} }{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } } \: \: }} \\\end{lgathered}

=

2+

3

2−

3

×

2−

3

2−

3

\begin{lgathered}\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \: \: \: \sqrt{ \dfrac{(2 - \sqrt{3} )^{2} }{ {(2)}^{2} - ( \sqrt{3})^{2} } } \: \: }} \\\end{lgathered}

=

(2)

2

−(

3

)

2

(2−

3

)

2

\begin{lgathered}\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \: \: \: \sqrt{ \dfrac{(2 - \sqrt{3} )^{2} }{4 - 3}} \: \: }} \\\end{lgathered}

=

4−3

(2−

3

)

2

\begin{lgathered}\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \: \: \: \sqrt{ {(2 - \sqrt{3} )^{2} }} \: \: }} \\\end{lgathered}

=

(2−

3

)

2

\begin{lgathered}\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \pm (2 - \sqrt{3}) }} \\\end{lgathered}

=±(2−

3

)

• Now put the value of { \bold{ \: \sqrt{3} \: \: - }}

3

−

\begin{lgathered}\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \pm \: (2 - 1.732) }} \\\end{lgathered}

=±(2−1.732)

\begin{lgathered}\\ { \bold{ \: \: \: \: \: \: \: \: = \: \: \pm \: (0.268) }} \\\end{lgathered}

=±(0.268)

Hence , \begin{lgathered}{ \bold{ \: \: \sqrt{ \dfrac{2 - \sqrt{3} }{2 + \sqrt{3} } } = \: \: \pm \: (0.268) \: \: }} \\\end{lgathered}

2+

3

2−

3

=±(0.268)

\begin{lgathered}\\ \bigstar \: \: \large { \red{ \bold{ \underline{used \: \: formula} } : - }} \\\end{lgathered}

★

usedformula

:−

\begin{lgathered}\\ { \blue{ \bold{(1) \: \: \: (a - b)(a + b) = {a}^{2} - {b}^{2} }}} \\\end{lgathered}

(1)(a−b)(a+b)=a

2

−b

2

\begin{lgathered}\\ { \blue{ \bold{(2) \: \: \: {(a -b)}^{2} = {a}^{2} + {b}^{2} - 2ab }}} \\\end{lgathered}

(2)(a−b)

2

=a

2

+b

2

−2ab