if x=3√2+√3 then x^3+1/x^3if the answer will correct I will mark u as brainlist About the author Margaret
Answer: [tex]\sf{x^{3} + \dfrac{1}{x^{3} } = \dfrac {91152\sqrt{2}+ 64106\sqrt{3}}{1125} }[/tex] Step-by-step explanation: Given: [tex]\sf{x = 3\sqrt{2} + \sqrt{3} }[/tex] To find: [tex]\sf{x^{3} + \dfrac{1}{x^{3} } = \; ?}[/tex] Solution: [tex]\sf{x^{3} + \dfrac{1}{x^{3} }}[/tex] Putting the value of x in the equation, [tex]\sf{ (3\sqrt{2} + \sqrt{3})^{3} + \dfrac{1}{ (3\sqrt{2} + \sqrt{3}) ^{3} }}[/tex] Simplifying,[tex]\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})\}} }[/tex] [tex]\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} \} } }[/tex] [tex]\hookrightarrow \sf{\{81\sqrt{2} + 57\sqrt{3}\} + \dfrac {1}{\{81\sqrt{2} + 57\sqrt{3}\} } }[/tex] Rationalizing the denominator, [tex]\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}\} }[/tex] [tex]\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac { \{81\sqrt{2} – 57\sqrt{3}\} }{ \{ (81\sqrt{2})^2 – (57\sqrt{3})^2 \} } }[/tex] [tex]\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac { \{81\sqrt{2} – 57\sqrt{3}\} }{ 13122 – 9747 } }[/tex] [tex]\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac { \{81\sqrt{2} – 57\sqrt{3}\} }{ 3,375 } }[/tex] Solving further, [tex]\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac { \not 3 (27\sqrt{2} – 19\sqrt{3}) }{ \not 3375 \;\; _ {_{\big {1125}}} } }[/tex] [tex]\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac {(27\sqrt{2} – 19\sqrt{3}) }{1125} }[/tex] [tex]\hookrightarrow \sf{\dfrac {91125\sqrt{2}+ 64125\sqrt{3} +27\sqrt{2} – 19\sqrt{3} }{1125} }[/tex] [tex]\hookrightarrow \sf{\dfrac {91152\sqrt{2}+ 64106\sqrt{3}}{1125} }[/tex] Hence, [tex]\sf{x^{3} + \dfrac{1}{x^{3} } = \dfrac {91152\sqrt{2}+ 64106\sqrt{3}}{1125} }[/tex] Formula used: (a+b)³ = a³ + b³ + 3a²b + 3ab² OR (a+b)³ = a³ + b³ + 3ab(a+b) (a+b) (a-b) = a² – b² Reply
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 Reply
Answer:
[tex]\sf{x^{3} + \dfrac{1}{x^{3} } = \dfrac {91152\sqrt{2}+ 64106\sqrt{3}}{1125} }[/tex]
Step-by-step explanation:
Given:
[tex]\sf{x = 3\sqrt{2} + \sqrt{3} }[/tex]
To find:
[tex]\sf{x^{3} + \dfrac{1}{x^{3} } = \; ?}[/tex]
Solution:
[tex]\sf{x^{3} + \dfrac{1}{x^{3} }}[/tex]
Putting the value of x in the equation,
[tex]\sf{ (3\sqrt{2} + \sqrt{3})^{3} + \dfrac{1}{ (3\sqrt{2} + \sqrt{3}) ^{3} }}[/tex]
Simplifying,[tex]\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})\}} }[/tex]
[tex]\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} \} } }[/tex]
[tex]\hookrightarrow \sf{\{81\sqrt{2} + 57\sqrt{3}\} + \dfrac {1}{\{81\sqrt{2} + 57\sqrt{3}\} } }[/tex]
Rationalizing the denominator,
[tex]\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}\} }[/tex]
[tex]\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac { \{81\sqrt{2} – 57\sqrt{3}\} }{ \{ (81\sqrt{2})^2 – (57\sqrt{3})^2 \} } }[/tex]
[tex]\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac { \{81\sqrt{2} – 57\sqrt{3}\} }{ 13122 – 9747 } }[/tex]
[tex]\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac { \{81\sqrt{2} – 57\sqrt{3}\} }{ 3,375 } }[/tex]
Solving further,
[tex]\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac { \not 3 (27\sqrt{2} – 19\sqrt{3}) }{ \not 3375 \;\; _ {_{\big {1125}}} } }[/tex]
[tex]\hookrightarrow \sf{ \{81\sqrt{2} + 57\sqrt{3}\} + \dfrac {(27\sqrt{2} – 19\sqrt{3}) }{1125} }[/tex]
[tex]\hookrightarrow \sf{\dfrac {91125\sqrt{2}+ 64125\sqrt{3} +27\sqrt{2} – 19\sqrt{3} }{1125} }[/tex]
[tex]\hookrightarrow \sf{\dfrac {91152\sqrt{2}+ 64106\sqrt{3}}{1125} }[/tex]
Hence,
[tex]\sf{x^{3} + \dfrac{1}{x^{3} } = \dfrac {91152\sqrt{2}+ 64106\sqrt{3}}{1125} }[/tex]
Formula used:
OR
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