If (x + 1/x ) = 5 sqrt2 , and x > 1, what is the value of (x⁶ - 1/(x⁶) ) ?

If (x + 1/x ) = 5 sqrt2 , and x > 1, what is the value of…

If (x + $\frac{1}{x}$ ) = 5 $\sqrt2$ , and x > 1, what is the value of (x⁶ - $\frac{1}{x^6}$ ) ?

Show answer & explanation

Correct answer: B

Explanation

The correct answer is option 2 i.e. 23030 $\sqrt23$ .

Given:

x + 1/x = 5√2

Formula used:

If x + 1/x = a then

⇒ x - 1/x = √a² - 4

⇒ (a + b)³ = a³ + b³ + 3ab(a + b)

⇒ (a - b)³ = a³ - b³ - 3ab(a - b)

Calculations:

⇒ x + 1/x = 5√2

Using equation (1), we get

⇒ x - 1/x = √46

Now using equation (2), we get

⇒ x³ + 1/x³ = 235√2

Similarly, using equation (3), we get

⇒ x³ - 1/x³ = 49√46

According to the question;

⇒ x⁶ - 1/x⁶ = (x³ + 1/x³)(x³ - 1/x³)

⇒ 235√2 × 49√46

⇒ 23030√23