Question

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




i want dis answer
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Answer 1

$\large\text{ \boxed{\bold{[Topic:\ Rationalization\ of\ the\ denominator]}} }$

Rationalization is effective in divisions. Let's take we are to calculate the value of $\large\text{ \dfrac{1}{\sqrt{2}} }$. If we only know that $\large\sqrt{2}\approx1.414$, it will consume an amount of time.

$\large\text{ \cdots\longrightarrow\dfrac{1}{\sqrt{2}}\approx\dfrac{1}{1.414}=? }$

But, since we know the definition of the square root, we can evaluate it faster. The method is called rationalization of the denominator.

$\large\text{ \cdots\longrightarrow\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}\approx\dfrac{1.414}{2}=0.707 }$

The above is a method for a rationalization for one irrational number. Now, to rationalize more terms, we use a polynomial identity.

$\large\text{ \cdots\longrightarrow\boxed{\begin{aligned}\bold{(a+b)(a-b)=a^{2}-b^{2}}\end{aligned}} }$

This will square each term, by definition of square root, the radicands are the result.

$\large\text{ \boxed{\bold{[Step\ 1.]}} }$

$\large\text{ \cdots\longrightarrow\text{(Given number)}=\dfrac{\sqrt{6}}{\sqrt{3}-\sqrt{2}} }$

As we see here the denominator is a difference of two irrational numbers. To resolve it, we simply multiply '1' to it.

if the non-zero denominator and numerator are equal the number is 1.

As we know, 1 is a multiplicative identity.

$\large\text{ \cdots\longrightarrow\text{(Given number)}=\dfrac{\sqrt{6}}{\sqrt{3}-\sqrt{2}}\times\dfrac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}} }$

$\large\text{ \boxed{\bold{[Step\ 2.]}} }$

Now we are given a polynomial identity of difference.

$\large\text{ \cdots\longrightarrow\boxed{\begin{aligned}\bold{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})=(\sqrt{3})^{2}-(\sqrt{2})^{2}}\end{aligned}} }$

So, -

$\large\text{ \cdots\longrightarrow\text{(Given number)}=\dfrac{\sqrt{6}(\sqrt{3}+\sqrt{2})}{(\sqrt{3})^{2}-(\sqrt{2})^{2}} }$

$\large\text{ \boxed{\bold{[Step\ 3.]}} }$

$\large\cdots\longrightarrow\text{(Given number)}=\sqrt{6}(\sqrt{3}+\sqrt{2})$

$\large\cdots\longrightarrow\text{(Given number)}=3\sqrt{2}+2\sqrt{3}$

Finally, the given approximation of irrational numbers gives it.

$\large\cdots\longrightarrow\text{(Given number)}\approx3\times1.414+2\times1.732$

$\large\cdots\longrightarrow\text{(Given number)}\approx4.242+3.464$

$\large\cdots\longrightarrow\text{(Given number)}\approx7.706$

$\large\text{ \boxed{\bold{[Final\ answer]}} }$

$\large\text{ \cdots\longrightarrow\boxed{\begin{aligned}\bold{\dfrac{\sqrt{6}}{\sqrt{3}-\sqrt{2}}\approx7.706}\end{aligned}} }$

Answer 2

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

➻ 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

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

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

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

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

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

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

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

➻ 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

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

$\bf \implies{\approx 7.706 }$❜❜

$\\ {\underline{\rule{300pt}{9pt}}}$

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

➻ 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:

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

$\\ {\underline{\rule{300pt}{9pt}}}$

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

➻ 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 ,

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

$\bf \implies{\approx 7.701 }$