Question

if
x = 2+√3
solve this Question
(1) ( x-1/x)3​

Answer 1

Answer:

24√3

Step-by-step explanation:

=> x = 2 + √3

=> 1/x = 1/(2 + √3)

Rationalize the RHS: multiply as well as divde RHS by 2 - √3:

=> 1/x = 1/(2 + √3) × (2 - √3)/(2 - √3)

=> 1/x = (2 - √3)/(2 + √3)(2 - √3)

=> 1/x = (2 - √3)/(2² - √3²)

=> 1/x = (2 - √3)/(4 - 3) = (2 - √3)/1

=> 1/x = 2 - √3

Subtract 1/x from x:

=> x - 1/x = (2 + √3) - (2 - √3)

=> x - 1/x = 2 + √3 - 2 + √3

=> x - 1/x = 2√3

Thus,

=> (x - 1/x)³ = (2√3)³ =2³*√3*√3*√3= 24√3

Answer 2

Answer :-

• Correct value = 24√3.

Step by step explanation :-

Given :-

• x = 2+√3.

To find :-

• Appropriate value of (x-1/x)³.

Concept :-

Here, the application of few identities have been implemented to find the value of required solution.

Rationalising is also done in order to simplify the given value.

Solution :-

⠀⠀⠀⠀⠀⠀⠀⠀⠀Given, x = 2+√3.

On equating x with 1/x :-

⠀⠀⠀⠀⠀⠀⠀⠀⠀1/2+√3 = 1/x

• Rationalising the given denominator.
• For rationalising, we'll simply multiply 2-√3 by both the sides.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ [1/2+√3]×[(2-√3)/(2-√3)]

• By using identity, (a+b)(a-b)=a²-b².

⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ [2-√3]/[(2)²-(√3)²]

⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ (2-√3)/(4-3)

⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ 2-√3.

On subtracting the value of 1/x from the x :-

⠀⠀⠀⠀⠀⠀⠀⠀⠀(2+√3)-(2-√3)

• Opening the brackets.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ 2+√3-2+√3.

• On cancelling 2 and -2.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ 2√3.

Obtaining the answer, i.e (x-1/x)³ :-

⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ (2√3)³

⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ 2×2×2×√3×√3×√3.

⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ 8 × 3√3

⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ 24√3 ✓

Hence,

• Appropriate value of (x-1/x)³ = 24√3.

⠀⠀⠀⠀⠀⠀⠀

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