Question

If x = 5 root2 -7 find the value of x^ 3 - 1/x^3​

Answer 1

Answer:

Step-by-step explanation:

Given that,

x = 5√2 - 7

1/ x = 1 / (5√2 - 7 )

Now rationalize the denominator

1 / x = 1 / ( 5√2 - 7 ) × (5√2 + 7) ( 5√2 + 7)

= 1 × ( 5√2 + 7 ) / ( 5√2 - 7 ) ( 5√2 + 7 )

= 5√2 + 7 / [ (5)^2 (√2 )^2 - ( 7 )^2 ] ( ∵ ( a+b ) (a -b ) = a ^2 - b ^2 )

= 5√2 + 7 / ( 25 × 2 - 49 ) ( ∵ root and square 2 are cancelled )

= 5√2 + 7 / ( 50 - 49 )

= 5√2 + 7 / ( 1 )

= 5√2 + 7

∴ 1 / x = 5√2 + 7

Now we find x^3 and 1 / x ^3

x^3 = ( 5√2 - 7 )^3

= ( 5√2 )^3 - 3 × ( 5√2 )^2 × 7 + 3 × 5√2 × ( 7 )^2 - ( 7 )^3

= [ ( 5 )^3 × ( √2 )^2 × ( √2 ) ] - 3 × [ ( 5 )^2 × ( √2 )^2 ] × 7 + 15√2 × 49 - 343

= 125 × 2 × √2 - 3 × 25 × 2 × 7 + 735√2 - 343

= 250√2 - 1020 + 735√2 - 343

= 985√2 - 1363

1 / ( x )^3 = ( 5√2 + 7 )^3

= (5√2)^3 + 3 × (5√2)^2 × 7 + 3 × 5√2 × ( 7 )^2 + ( 7 )^3

= [ (5)^3 × (√2)^3 ] + 3 × [ (5)^2 × (√2 )^2 ] × 7 + 15√2 × 49 + 343

= [ 125 × (√2)^2 × (√2) ] + 3 × 25 × 2 × 7 + 735√2 + 343

= 125 ×2 × √2 + 1020 + 735√2 + 343

= 250√2 + 735√2 + 1020 + 343

= 935√2 + 1363

∴ x ^3 - 1 / x ^3 = ( 935√2 - 1363 ) - ( 935√2 + 1363 )

= 935√2 - 1363 - 935√2 - 1363

= - 1363 - 1363 ( ∵ both +935√2 and -935√2 are cancelled )

= - 2726 is the answer. ( ∵both are added but symbol is same )

Answer 2

Step-by-step explanation:

$x = 5 \sqrt{2} - 7 \\ \\ \frac{1}{x} = \frac{1}{5 \sqrt{2} - 7 } \\ \frac{1}{x} = \frac{5 \sqrt{2} + 7}{50 - 49} \\ \frac{1}{x} = 5 \sqrt{2} + 7 \\ \\ x - \frac{1}{x} = 5 \sqrt{2} - 7 - 5 \sqrt{2} - 7 \\ = - 14 \\ \\ {x}^{3} - \frac{1}{ {x}^{3} } = ( {x - \frac{1}{x} })^{3} + 3(x - \frac{1}{x} ) \\ = ( { - 14})^{3} + 3( - 14) \\ = - 2744 - 52 \\ = - 2796$