Question

If z^4+1/z^4=47, find the value of z^3+1/z^3.

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Answer 1

Given : z⁴ + 1/z⁴ = 47

To find : z³ + 1/z³ = ?

Solution :

Now, z⁴ + 1/z⁴ = 47

⇒ (z² + 1/z²)² - (2 * z² * 1/z²) = 47

⇒ (z² + 1/z²)² - 2 = 47

⇒ (z² + 1/z²)² = 47 + 2 = 49

⇒ z² + 1/z² = 7 (positive value only)

⇒ (z + 1/z)² - (2 * z * 1/z) = 7

⇒ (z + 1/z)² - 2 = 7

⇒ (z + 1/z)² = 7 + 2 = 9

⇒ z + 1/z = 3 (positive value only)

∴ z³ + 1/z³

= (z + 1/z) (z² + 1/z² - 1)

= 3 (7 - 1)

= 3 * 6

= 18

⇒ z³ + 1/z³ = 18