# if x=3√2+√3 then x^3+1/x^3if the answer will correct I will mark u as brainlist ​

if x=3√2+√3 then x^3+1/x^3
if the answer will correct I will mark u as brainlist ​

### 2 thoughts on “if x=3√2+√3 then x^3+1/x^3<br />if the answer will correct I will mark u as brainlist ​”

$$\sf{x^{3} + \dfrac{1}{x^{3} } = \dfrac {91152\sqrt{2}+ 64106\sqrt{3}}{1125} }$$

Step-by-step explanation:

Given:

$$\sf{x = 3\sqrt{2} + \sqrt{3} }$$

To find:

$$\sf{x^{3} + \dfrac{1}{x^{3} } = \; ?}$$

Solution:

$$\sf{x^{3} + \dfrac{1}{x^{3} }}$$

Putting the value of x in the equation,

$$\sf{ (3\sqrt{2} + \sqrt{3})^{3} + \dfrac{1}{ (3\sqrt{2} + \sqrt{3}) ^{3} }}$$

Simplifying,$$\hookrightarrow \sf{\{(3\sqrt{2})^{3} + (\sqrt{3})^{3} + 3 \times (3\sqrt{2})^{2} \times (\sqrt{3})\} + \dfrac {1}{\{(3\sqrt{2})^{3} + (\sqrt{3})^{3} + 3 \times (3\sqrt{2})^{2} \times (\sqrt{3})\}} }$$

$$\hookrightarrow \sf{\{54\sqrt{2} + 54\sqrt{3} + 27\sqrt{2} + 3\sqrt{3} \} + \dfrac {1}{\{54\sqrt{2} + 54\sqrt{3} + 27\sqrt{2} + 3\sqrt{3} \} } }$$

$$\hookrightarrow \sf{\{81\sqrt{2} + 57\sqrt{3}\} + \dfrac {1}{\{81\sqrt{2} + 57\sqrt{3}\} } }$$

Rationalizing the denominator,

$$\hookrightarrow \sf{\{81\sqrt{2} + 57\sqrt{3}\} + \dfrac {1}{\{81\sqrt{2} + 57\sqrt{3}\} } \times \dfrac {\{81\sqrt{2} – 57\sqrt{3}\}}{\{81\sqrt{2} – 57\sqrt{3}\} }$$

$$\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac { \{81\sqrt{2} – 57\sqrt{3}\} }{ \{ (81\sqrt{2})^2 – (57\sqrt{3})^2 \} } }$$

$$\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac { \{81\sqrt{2} – 57\sqrt{3}\} }{ 13122 – 9747 } }$$

$$\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac { \{81\sqrt{2} – 57\sqrt{3}\} }{ 3,375 } }$$

Solving further,

$$\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac { \not 3 (27\sqrt{2} – 19\sqrt{3}) }{ \not 3375 \;\; _ {_{\big {1125}}} } }$$

$$\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac {(27\sqrt{2} – 19\sqrt{3}) }{1125} }$$

$$\hookrightarrow \sf{\dfrac {91125\sqrt{2}+ 64125\sqrt{3} +27\sqrt{2} – 19\sqrt{3} }{1125} }$$

$$\hookrightarrow \sf{\dfrac {91152\sqrt{2}+ 64106\sqrt{3}}{1125} }$$

Hence,

$$\sf{x^{3} + \dfrac{1}{x^{3} } = \dfrac {91152\sqrt{2}+ 64106\sqrt{3}}{1125} }$$

Formula used:

• (a+b)³ = a³ + b³ + 3a²b + 3ab²

OR

• (a+b)³ = a³ + b³ + 3ab(a+b)
• (a+b) (a-b) = a² – b²
2. ## Solution!!

x = 3√2 + √3

x³ + 1/x³ = (3√2 + √3)³ + 1/(3√2 + √3)³

= (54√2 + 54√3 + 27√2 + 3√3) + 1/(3√2 + √3)³

= (54√2 + 54√3 + 27√2 + 3√3) + 1/(54√2 + 54√3 + 27√2 + 3√3)

= (81√2 + 57√3) + 1/(54√2 + 54√3 + 27√2 + 3√3)

= (81√2 + 57√3) + 1/(81√2 + 57√3)

= (81√2 + 58√3) + 1/(81√2 + 57√3) × (81√2 – 57√3)/(81√2 – 57√3)

= (81√2 + 57√3) + [1(81√2 – 57√3)/(81√2 + 57√3)(81√2 – 57√3)]

= (81√2 + 57√3) + [(81√2 – 57√3)/(81√2)² – (57√3)²]

= (81√2 + 57√3) + (81√2 – 57√3)/(13122 – 9747)

= (81√2 + 57√3) + (81√2 – 57√3)/3375

= (81√2 + 57√3) + [3(27√2 – 19√3)/3375]

= (81√2 + 57√3) + (27√2 – 19√3)/1125

= (91125√2 + 64125√3 + 27√2 – 19√3)/1125

= (91152√2 + 64106√3)/1125