Question

If z²+1/z²=14, find the value of z³+1/z³ by taking only the positive value of z+1/z

Answer 1

Answer:

52.

Step-by-step explanation:

Using the identity a^3 + b^3 = (a + b)(a^2 - ab + b^3):

z^3 + 1/z^3 = (z + 1/z)(z^2 - z *1/z + 1/z^2)

= (z + 1/z) (z^2 - 1 + 1/z^2)

= (z + 1/z) (14 - 1 )

= 13(z + 1/z).

Now using the identity a^2 + b^2 = (a + b)^2 - 2ab:

z^2 + 1/z^2 = ( z + 1/z)^2 - 2(z * 1/z)  = 14 So:

( z + 1/z)^2 - 2 = 14

( z + 1/z)^2 = 16

Taking z + 1/z = 4 we therefore have:

z^3 + 1/z^3 =  13 * 4 = 52.

Answer 2

THE ANSWER IS 52

Given z²+1/z² =14

using above equation,

z²+1/z² = 14

add 2 on both side,

z² + 1/z² + 2 = 14 + 2

the LHS is whole square of z + 1/z

so we can write it

(z + 1/z)² = 16z + 1/z = 4

Now we know,

(a + b)³ = a³ + b³ + 3ab(a+b)

here, a = z and b = 1/z,

substituting a and b in equation we will get,

(z + 1/z)³ = z³ + 1/z³ + 3z/z( z+ 1/z)

4³ = z³ + 1/z³ + 3×4

z³+ 1/z³ = 64-12

z³ + 1/z³ = 52