Math, asked by gururaj272700, 10 months ago

If x= 2+√3, find the value of x³+1/x³

Answers

Answered by singhvardan256
0

Answer:

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

3

To find ;

x {}^{3} + \frac{1}{x {}^{3} }x

3

+

x

3

1

Solution :

\begin{lgathered}x = 2 + \sqrt{3} \\ \\ \frac{1}{x} = \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ \\ \frac{1}{x} = \frac{2 - \sqrt{3} }{(2) {}^{2} - (\sqrt{3} ) {}^{2} } \\ \\ \frac{1}{x} = \frac{2 - \sqrt{3} }{4 - 3} \\ \\ \frac{1}{x} = 2 - \sqrt{3}\end{lgathered}

x=2+

3

x

1

=

2+

3

1

×

2−

3

2−

3

x

1

=

(2)

2

−(

3

)

2

2−

3

x

1

=

4−3

2−

3

x

1

=2−

3

Now,

\begin{lgathered}x + \frac{1}{x} \\ \\ \implies2 + \cancel{\sqrt{3}}+ 2 - \cancel {\sqrt{3}} \\ \\ \implies2 + 2 \\ \\ \implies4\end{lgathered}

x+

x

1

⟹2+

3

+2−

3

⟹2+2

⟹4

So, on cubing both sides, we have

\begin{lgathered}(x + \frac{1}{x} ) {}^{3} = (4){}^{3} \\ \\ x{}^{3} + \frac{1}{x{}^{3}} + 3(x + \frac{1}{x} ) = 64 \\ \\ x{}^{3} + \frac{1}{x{}^{3}} +3 \times 4 = 64 \\ \\ x{}^{3} + \frac{1}{x{}^{3}} +12 = 64 \\ \\ x{}^{3} + \frac{1}{x{}^{3}} = 64 - 12 \\ \\ x{}^{3} + \frac{1}{x{}^{3}} = 64 - 12 \\ \\ \boxed{ \bold{ x{}^{3} + \frac{1}{x{}^{3}} = 52}}\end{lgathered}

(x+

x

1

)

3

=(4)

3

x

3

+

x

3

1

+3(x+

x

1

)=64

x

3

+

x

3

1

+3×4=64

x

3

+

x

3

1

+12=64

x

3

+

x

3

1

=64−12

x

3

+

x

3

1

=64−12

x

3

+

x

3

1

=52

_______________________

Thanks for the questions.

Answered by anindyaadhikari13
2

Given,

  • x = 2 +  \sqrt{3}

To find:-

  • The value of x

Solution:-

x = 2 +  \sqrt{3}

 \implies  \frac{1}{x}  =  \frac{1}{2 +  \sqrt{3} }

 =  \frac{1}{2 +  \sqrt{3} }  \times  \frac{2 -  \sqrt{3} }{2 -  \sqrt{3} }

 =  \frac{2 -  \sqrt{3} }{ {(2)}^{2}-  {( \sqrt{3} )}^{2}  }

 =  \frac{2 -  \sqrt{3} }{4 - 3}

 =  \frac{2 -  \sqrt{3} }{1}

 = 2 -  \sqrt{3}

So,

 {x}^{3}  +  \frac{1}{ {x}^{3} }

 =  {(x +  \frac{1}{x} )}^{3}  - 3 \times x \times  \frac{1}{x} (x +  \frac{1}{x} )

Now,

x +  \frac{1}{x}

 = 2 +  \sqrt{3}  + 2 -  \sqrt{3}

 = 4

So,

 {(x +  \frac{1}{x} )}^{3}  - 3 \times x \times  \frac{1}{x} (x +  \frac{1}{x} )

 =  {4}^{3}  - 3 \times 4

 = 64 - 12

 = 52

Answer:-

  •  {x}^{3}  +  \frac{1}{ {x}^{3} }  = 52
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