Question

if x =2+√3 find the value of x^3+1/x^3
please send me the pic of the solution ​

Answer 1

52

Step-by-step explanation:

Given:

x = 2 + √3

To find:

$\textsf{The value of } \: \: {x}^{3} + \dfrac{1}{ {x}^{3} }$

How to Solve:

1. We the value of (x + 1/x)

2. We have to find the value of 1/x

3. Algebra Formula

[a³ + b³ = (a+b)³- 3ab (a+b)] ---(2)

[a² - b² = (a - b)(a + b)] ---(1)

will be used.

4. Rationalizing will also required.

Solution:

x = 2 + √3

$\textsf{So,} \: \dfrac{1}{x} = \dfrac{1}{2 + \sqrt{3} } \\ \\ = \frac{(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})}$

Rationalizing the Denominator,

$= \frac{(2 - \sqrt{3})}{ {(2)}^{2} - {( \sqrt{3})}^{2} } \: \: [ \text{Using 1}]\\ \\ = \frac{2 - \sqrt{3} }{4 - 3} \\ \\ \therefore \: \: \: \frac{1}{x} = 2 - \sqrt{3}$

Now,

$x + \frac{1}{x} =( 2 + \sqrt{3} ) + (2 - \sqrt{3} ) \\ \\ = 2 + 2 + \sqrt{3} - \sqrt{3} \\ \\ = 4$

Now,

${x}^{3} + \frac{1}{ {x}^{3} } \: \:[ \text{Using 2}] \: \\ \\ = (x + \frac{1}{x} ) {}^{3} - 3.x. \frac{1}{x} (x + \frac{1}{x} ) \\ \\ = {(4)}^{3} - 3(4) \\ \\ = 64 - 12 \\ \\ = 52$

Therefore, The required answer is 52 .