Question

if x = √3+1/(√3-1) + √3-1/(√3+1) + √3-2/(√3+2) , find the value of x² + (39/x)².​

Answer 1

Step-by-step explanation:

As per the information provided in the question, We have :

• x = √3+1/(√3-1) + √3-1/(√3+1) + √3-2/(√3+2)

We are asked to find the value of x² + (39/x)².

In order to find the value of x² + (39/x), We need to rationalize the value of x. As the value of x is a very large number, We will divide it into three parts, And rationalize it. After Rationalising the whole thing (The value of x), We will put the it in the given Equation x² + (39/x)².

It will be divided into three parts i.e √3+1/(√3-1), √3-1/(√3+1) & √3-2/(√3+2).

First part!

$\longmapsto \rm \dfrac{ \sqrt{3} + 1}{ \sqrt{3} - 1}$

Multiplying it with its rationalising factor,

$\longmapsto \rm \dfrac{ \sqrt{3} + 1}{ \sqrt{3} - 1} \times \dfrac{ \sqrt{3} + 1 }{\sqrt{3} + 1}$

Rearranging the terms.

$\longmapsto \rm \dfrac{ {( \sqrt{3} + 1)}^{2} }{ {(\sqrt{3} - 1) }^{2} }$

$\longmapsto \rm \dfrac{ { (\sqrt{3}) }^{2} + ( {1)}^{2} + 2 \sqrt{3} (1) }{ {(\sqrt{3} - 1) }^{2} }$

$\longmapsto \rm \dfrac{ {3}+ 1 + 2 \sqrt{3} }{ {(\sqrt{3} - 1) }^{2} }$

$\longmapsto \rm \dfrac{ 4 + 2 \sqrt{3} }{ 3 - 1 }$

$\longmapsto \rm \dfrac{ 2(2 + \sqrt{3} )}{ 2 }$

$\longmapsto \rm 2 + \sqrt{3}$

Second part!

$\longmapsto \rm \dfrac{ \sqrt{3} - 1}{ \sqrt{3} + 1}$

Multiplying it with its rationalising factor,

$\longmapsto \rm \dfrac{ \sqrt{3} - 1}{ \sqrt{3} + 1} \times \dfrac{ \sqrt{3} - 1}{ \sqrt{3} - 1}$

Rearranging the terms.

$\longmapsto \rm \dfrac{ {( \sqrt{3} - 1)}^{2} }{ {(\sqrt{3} - 1) }^{2} }$

$\longmapsto \rm \dfrac{ {( \sqrt{3} ) + ( {1)}^{2} } - 2 \sqrt{3} }{ 3 - 1 }$

$\longmapsto \rm \dfrac{{3 + 1} - 2 \sqrt{3} }{ 3 - 1 }$

$\longmapsto \rm \dfrac{2({2} - \sqrt{3}) }{ 2 }$

$\longmapsto \rm{ {2} - \sqrt{3} }$

Third part!

$\longmapsto \rm \dfrac{ \sqrt{3} - 2}{ \sqrt{3} + 2}$

$\longmapsto \rm \dfrac{ \sqrt{3} - 2}{ \sqrt{3} + 2} \times \dfrac{ \sqrt{3} + 2}{ \sqrt{3} + 2}$

$\longmapsto \rm \dfrac{ (\sqrt{3} - 2)^{2} }{ (\sqrt{3} + 2)(\sqrt{3} - 2)}$

$\longmapsto \rm \dfrac{ 7 - 4 \sqrt{3} }{ - 1}$

$\longmapsto \rm - ( {7 - 4 \sqrt{3} ) }$

Adding all the three parts :

$\longmapsto \rm 2 + \sqrt{3} + 2 - \sqrt{3} + ( - (7 - 4 \sqrt{3} ))$

$\longmapsto \rm 2 + \sqrt{3} + 2 - \sqrt{3} - 7 + 4 \sqrt{3}$

$\longmapsto \rm 2 + 2 - 7 + 4 \sqrt{3}$

$\longmapsto \rm 4 - 7 + 4 \sqrt{3}$

$\longmapsto \rm 4 \sqrt{3} - 3$

∴ The value of x after Rationalizing is 4√3 - 3.

Putting the value of x in the given equation :

$\longmapsto \rm { x}^{2} +{ \bigg ( \dfrac{39}{x} \bigg)}^{2}$

$\longmapsto \rm { (4 \sqrt{3} - 3)}^{2} +{ \bigg ( \dfrac{39}{4 \sqrt{3} - 3} \bigg)}^{2}$

$\longmapsto \rm 57 - 24 \sqrt{3} +{ \bigg ( \dfrac{39}{4 \sqrt{3} - 3} \bigg)}^{2}$

$\longmapsto \rm 57 - 24 \sqrt{3} +{ \dfrac{ {(39)}^{2} \div 3 }{ {{(4 \sqrt{3} - 3}) }^{2} } }$

$\longmapsto \rm 57 - 24 \sqrt{3} +{ \dfrac{ 507}{ 19 - 8 \sqrt{3} }}$

$\longmapsto \rm 57 - 24 \sqrt{3} +3(19 - 8 \sqrt{3} )$

$\longmapsto \rm 57 - 24 \sqrt{3} +57 - 24 \sqrt{3}$

$\longmapsto \rm 114 - 48 \sqrt{3}$

∴ The value of x² + (39/x)² is 114 - 48√3.