Question

plz, solve this question. Q.no.9, 10, 11

Answer 1

this is the answer by using identities

Answer 2

9.

$x^2 + \frac{1}{x^2} = 102 \\ \\ x^2 + \frac{1}{x^2} - 2 = 102 - 2 \\ \\ = (x - \frac{1}{x})^2 = 100 \\ \\ x - \frac{1}{x} = \sqrt{100} = 10 \\ \\ \\ (x - \frac{1}{x})^3 = 10^3 \\ \\ = x^3 - 3x + \frac{3}{x} - \frac{1}{x^3} = 1000 \\ \\ = x^3 - 3(x - \frac{1}{x}) - \frac{1}{x^3} = 1000 \\ \\ = x^3 - 3 \times 10 - \frac{1}{x^3} = 1000 \\ \\ = x^3 - 30 - \frac{1}{x^3} = 1000 \\ \\ \\ x^3 - \frac{1}{x^3} = 1000 + 30 = \bold{1030}$

10.

$x^2 + \frac{1}{x^2} = 14 \\ \\ x^2 + \frac{1}{x^2} + 2 = 14 + 2 \\ \\ = (x + \frac{1}{x})^2 = 16 \\ \\ x + \frac{1}{x} = \sqrt{16} = 4 \\ \\ \\ (x + \frac{1}{x})^3 = 4^3 \\ \\ = x^3 + 3x + \frac{3}{x} + \frac{1}{x^3} = 64 \\ \\ = x^3 + 3(x + \frac{1}{x}) + \frac{1}{x^3} = 64 \\ \\ = x^3 + 3 \times 4 + \frac{1}{x^3} = 64 \\ \\ = x^3 + 12 + \frac{1}{x^3} = 64 \\ \\ \\ x^3 + \frac{1}{x^3} = 64 - 12 = \bold{52}$

11.

$x^4 + \frac{1}{x^4} = 47 \\ \\ x^4 + \frac{1}{x^4} + 2 = 47 + 2 \\ \\ = (x^2 + \frac{1}{x^2})^2 = 49 \\ \\ x^2 + \frac{1}{x^2} = \sqrt{49} = 7 \\ \\ x^2 + \frac{1}{x^2} + 2 = 7 + 2 \\ \\ = (x + \frac{1}{x})^2 = 9 \\ \\ x + \frac{1}{x} = \sqrt{9} = 3 \\ \\ \\ (x + \frac{1}{x})^3 = 3^3 \\ \\ = x^3 + 3x + \frac{3}{x} + \frac{1}{x^3} = 27 \\ \\ = x^3 + 3(x + \frac{1}{x}) + \frac{1}{x^3} = 27 \\ \\ = x^3 + 3 \times 3 + \frac{1}{x^3} = 27 \\ \\ = x^3 + 9 + \frac{1}{x^3} = 27$

$\\ \\ \\ x^3 + \frac{1}{x^3} = 27 - 9 = \bold{18}$

Also we can apply the below formula:

$If\ x^2 + \frac{1}{x^2} = n,\ then\ x^3 + \frac{1}{x^3} = (n - 1)\sqrt{n + 2} \\ \\ If\ x^2 + \frac{1}{x^2} = n,\ then\ x^3 - \frac{1}{x^3} = (n + 1)\sqrt{n - 2} \\ \\ If\ x^4 + \frac{1}{x^4} = n,\ then\ x^3 + \frac{1}{x^3} = (\sqrt{n + 2} - 1)\sqrt{\sqrt{n + 2} + 2}$

Let's check.

$9.\ (102 + 1)\sqrt{102 - 2} = 103\sqrt{100} = 103 \times 10 = \bold{1030} \\ \\ 10.\ (14 - 1)\sqrt{14 + 2} = 13\sqrt{16} = 13 \times 4 = \bold{52} \\ \\ 11.\ (\sqrt{47 + 2} - 1)\sqrt{\sqrt{47 + 2} + 2} \\ \\ = (\sqrt{49} - 1)\sqrt{\sqrt{49} + 2} \\ \\ = (7 - 1)\sqrt{7 + 2} = 6\sqrt{9} = 6 \times 3 = \bold{18}$

Hope this may be helpful.

Please mark my answer as the brainliest if this may be helpful.

Thank you. Have a nice day.