Question

If x+1/x =3 then find the value of x^6+1/x^6

Answer 1

Answer:

Step-by-step explanation:

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

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

$\\ \\ \\ (x + \frac{1}{x})^6 = 3^6 \\ \\ = x^6 + (6 \times x^5 \times \frac{1}{x}) + (15 \times x^4 \times \frac{1}{x^2}) + (20 \times x^3 \times \frac{1}{x^3}) + (15 \times x^2 \times \frac{1}{x^4}) + (6 \times x \times \frac{1}{x^5}) + \frac{1}{x^6} = 729 \\ \\ = x^6 + 6x^4 + 15x^2 + 20 + \frac{15}{x^2}$

Answer 2

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