Question

if x =root 5-root3, then find x^2+1/x^2 and x^3+1/x^3

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Answer 1

• x² + 1/x² = (20 - 3√15)/2

• x³ + 1/x³ = 63/4 * (√5 - √3)

EXPLANATION:

Given, x = √5 - √3

Then 1/x = 1/(√5 - √3)

= (√5 + √3)/{(√5 - √3) (√5 + √3)}

= (√5 + √3)/(5 - 3)

= (√5 + √3)/2

& x + 1/x = √5 - √3 + (√5 + √3)/2

= (2√5 - 2√3 + √5 + √3)/2

= (3√5 - √3)/2

Now, x² + 1/x²

= (x + 1/x)² - 2 * x * 1/x

= {(3√5 - √3)/2}² - 2

= (45 - 6√15 + 3)/4 - 2

= (48 - 6√15 - 8)/4

= (40 - 6√15)/4

= (20 - 3√15)/2

& x³ + 1/x³

= (x + 1/x) (x² + 1/x² - x * 1/x)

= (3√5 - √3)/2 * {(20 - 3√15)/2 - 1}

= (3√5 - √3)/2 * (20 - 3√15 - 2)/2

= (3√5 - √3)/2 * (18 - 3√15)/2

= (3√5 - √3)/2 * 3/2 * (6 - √15)

= 3/4 * (3√5 - √3) (6 - √15)

= 3/4 * (18√5 - 15√3 - 6√3 + 3√5)

= 3/4 * (21√5 - 21√3)

= 3/4 * 21 (√5 - √3)

= 63/4 * (√5 - √3)

Answer 2

Step-by-step explanation:

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