Question

if√2=1.414,√3=1.732, then find the value of√6by√3-√2

Answer 1

$\large \underline{\underline{\rm{ \bull \: Solution : - }}}$

➻ The given values are as follows :

$\rm{ \mapsto \: \sqrt{2} =1.414 }$

$\rm{ \mapsto \: \sqrt{3} =1.732}$

➻ What we have to find out :

We have to find the value of

$\rm{ \longrightarrow \:\dfrac{ \sqrt{6} }{ \sqrt{3} - \sqrt{2} } }$

$\overline{\rule{ 200pt}{2pt}}$

$\large \underline{\underline{\rm{ \bull \:Assume \: that : - }}}$

$\rm \leadsto{Let \: t= \dfrac{ \sqrt{6} }{ \sqrt{3} - \sqrt{2} } }$

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$\large \underline{\underline{\rm{ \bull \: Step \: by \: step \: solution : - }}}$

$\large \underline{\underline{\rm{ \bull \: Step \:1: - }}}$

➻ To make denominator an integer we have to first rationalize the given quantity .

➻On rationalizing, therefore we get:

$\rm \mapsto{\: t= \dfrac{ \sqrt{6} }{ \sqrt{3} - \sqrt{2} } \times \dfrac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} + \sqrt{2} } }$

$\rm \mapsto{\: t= \dfrac{ \sqrt{6} \: (\sqrt{3} + \sqrt{2}) }{ (\sqrt{3} - \sqrt{2} ) \: ( \sqrt{3} + \sqrt{2} )} }$

$\large \underline{\underline{\rm{ \bull We\:know \: that : - }}}$

$\rm \implies \: { {a}^{2} - {b}^{2} = (a + b)(a - b) }$

$\large \underline{\underline{\rm{ \bull \: Step \:2: - }}}$

➻ To get the answer use this identity and the given values

➻ By using this identity for the above equation we get :-

$\rm \mapsto{\: t= \dfrac{ \sqrt{6} \: (\sqrt{3} + \sqrt{2}) }{( { \sqrt{3} \: ) }^{2} - {( \sqrt{2} \: )}^{2} } }$

$\rm \mapsto{\: t= \dfrac{ \sqrt{6} \: (\sqrt{3} + \sqrt{2}) }{ { 3 - 2 }} }$

$\rm \mapsto{\: t= \dfrac{ \sqrt{6} \: (\sqrt{3} + \sqrt{2}) }{ { 1}} }$

$\rm \mapsto{\: t= \sqrt{6} \: ( \sqrt{3} + \sqrt{2} ) }$

$\rm \mapsto{\: t= 3 \sqrt{2} + 2 \sqrt{3} }$

$\large \underline{\underline{\rm{ \bull \: Step \:3: - }}}$

➻ Substitute the obtained values

➻Therefore by substituting the given values we get :-

$\rm \longrightarrow{\: t = 3 \times 1.414 + 2 \times 1.732 }$

$\rm \longrightarrow{\: t = 4.242 + 3.464 }$

$\rm \implies{\approx 7.706 }$

$\large \underline{\underline{\rm{ \bull Therefore : - }}}$

➻ ❛❛ The value of

$\rm{ \longrightarrow \:\dfrac{ \sqrt{6} }{ \sqrt{3} - \sqrt{2} \: } \: is }$

$\bf \implies{\approx 7.706 }$❜❜

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$\large \underline{\underline{\rm{ \bull \: Learn \: More : - }}}$

➻ The method of converting the surd denominator into an integer by multiplying the denominator and the numerator by the conjugate of the denominator is known as Rationalization of the expression.

➻ The above identity is given by:

$\rm \implies \: { {a}^{2} - {b}^{2} = (a + b)(a - b) }$

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$\large \underline{\underline{\rm{ \bull \: Note : - }}}$

➻ Let's check the obtained answer is correct or not by directly substituting the given values and then further solving the fraction to get the required value .

$\rm{ \mapsto \:\dfrac{ \sqrt{6} }{ \sqrt{3} - \sqrt{2} \: } \: }$

➻ Now by substituting the given values we get :

$\rm{ \longrightarrow \:\dfrac{ \sqrt{6} }{ 1.732- 1.414 } }$

$\rm{ \longrightarrow \:\dfrac{ \sqrt{6} }{ 0.318 } }$

➻ As value of √6 is 2.449 So ,

$\rm{ \longrightarrow \:\dfrac{2.449}{ 0.318 } }$

$\bf \implies{\approx 7.701 }$

Answer 2

Answer:

⇒

2

=1.414,

3

=1.732

∴

3

3

−2

2

4

+

3

3

+2

2

3

⇒

3∗1.732−2∗1.414

4

+

3∗1.732+2∗1.414

3

⇒

5.196−2.828

4

+

5.196+2.828

3

⇒

2.368

4

+

8.024

3

⇒1.960+0.373=2.063

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Step-by-step explanation:

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