Question

Hey!!

If , x=4+√15 then find the value of x³- 1/x³

Answer 1

Answer :-

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Given :-

x = ( 4 + √15 )

To find :-

$\mathsf{The \: value \:of\: x^{3} - \frac{1}{x^{3}}}$

Salutation :-

Given , x = ( 4 + √15 )

Then ,

1 / x = 1 / ( 4 + √15 )

= 1 / ( 4 + √15 ) × 1

= 1 / ( 4 + √15 ) × ( 4 - √15 ) / ( 4 - √15 )

= ( 4 - √15 ) / { ( 4 + √15 ) × ( 4 - √15 ) }

= ( 4 - √15 ) / ( 4 )² - ( √15 )²

[ • As we know , a² - b² = ( a + b ) ( a - b ) ]

= ( 4 - √15 ) / ( 16 - 15 )

= ( 4 - √15 ) / 1

= ( 4 - √15 )

So ,

$\mathsf{x - \frac{1}{x}}$

= ( 4 + √15 ) - ( 4 - √15 )

= 4 + √15 - 4 + √15

= + 2√15

Therefore ,

$\mathsf{ x^{3} - \frac{1}{x^{3}}}$

= ( x - 1 / x )³ + 3 × x × 1 / x ( x - 1 / x ) [ • Using identity ]

= ( 2√15 )³ + 3 × 2√15

= 8 × 15√15 + 6√15

= 120√15 + 6√15

= 126√15                     [ ★ Required answer ]

•°• The value of x³ - 1 / x³ is 126√15.

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Answer 2

$<b>Heya mate. (^_-). Solution below.$
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• x = 4 + √15



• Then,

(1/x) = (1/4 + √15)

Multiplying with the conjugate -

(1/x)

= (1/4 + √15) × (4 - √15/4 - √15)

= (1 × 4 - √15)/(4² - √15²)

= (4 - √15)/(16 - 15)

= (4 - √15)/1

= 4 - √15

• So, (1/x) = 4 - √15



• Now, x - (1/x)

= 4 + √15 - (4 - √15)

= 4 + √15 - 4 + √15

= 2√15

• So, x - (1/x) = 2√15



• Now, cubing both sides

[x - (1/x)]³ = [2√15]³


•Using identity : (a - b)³ = a³ - b³ - 3ab(a - b)


=> x³ - (1/x³) - (3 × x × (1/x)) (x - (1/x)) = 2³ × √15³

=> x³ - (1/x³) - 3 (2√15) = 8 × 15√15

=> x³ - (1/x³) - 6√15 = 120√15

=> x³ - (1/x³) = 120√15 + 6√15

=> x³ - (1/x³) = 126√15.....$\boxed{\mathbb{\red{\bold{ANSWER!!}}}}$


$<marquee>Hence, your answer is 126√15.</marquee>$
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Thank you.. ;-)