Question

ACCURATE ANSWERS EXPECTED

$if \: x = 2 + \sqrt{3} \\ find \: \: the \ \: value \: \: of \: {x}^{3} + \frac{1}{x {}^{3} }$

Answer 1

Hey !!!

x = 2 +√3 --------1)

by using rationalising process ...

1 /x =1/ 2 + √3 × 2 -√3 /2-√3

=> 2- √3 / (2)²- (√3)²

=> 2 -√3 /4-3


hence , 1 /x = 2-√3-------2)

if we adding equation 1 and 2 we got

x + 1/x =2 + √3 + 2- √3 = 4 -----3

by using (a + b)² = a² + b² + 2ab

now , (x + 1/x )² = x² +( 1/x)² + 2x ×1/x

(4 )² = x² + 1/x² +2

16-2 =x² + 1/x² ----------4)

Now, as we know that ...

a³ + b³ ={( a+ b)( a² +b² -ab)}

so like that ,

x³+ 1/x³ = (x+1/x)(x² + 1/x² -x×1/x)}

x³ +1/x³ ={ (4) (14 -1 )} ▶[ from equation 3 ,and 4 ]


x³ + 1/x³ = 4×13

x³ +1/x³ = 52 Answer. ..

hope it helps !!!

#rajukumar111

Answer 2

HELLO DEAR,

X = 2+√3


in cubing both side

we get,

we know that:-

(a+b)³ = a³+b³+3ab(a+b)

x³ = 2³+√3³+3×2×√3(2+√3)

=> 8+3√3 +6√3(2+√3)

=> 8+3√3+12√3+18

=> (26+15√3)


${x}^{3} + \frac{1}{ {x}^{3} } \\ \\ = > 26 + 15\sqrt{3} + \frac{1}{26 + 15 \sqrt{3} } \\ \\ = > 26 + 15\sqrt{3} + \frac{( 26 - 15 \sqrt{3} )}{(26 + 15 \sqrt{3}) ( 26 - 15 \sqrt{3}) } \\ \\ = > 26 + 15 \sqrt{3} + \frac{26 - 15 \sqrt{3} }{ 676 - 675} \\ \\ = > 26 + 15 \sqrt{3} + \frac{26 - 15 \sqrt{3} }{1} \\ \\ = > 26 + 15 \sqrt{3} + 26 - 15\sqrt{3} \\ \\ 26 + 26 = 52$
I HOPE ITS HELP YOU DEAR,
THANKS