Question

If x = 3+2√2 find x ​4​ + 1/x​4​.

Answer 1

Step-by-step explanation:

(3+2√2)4 + 1/(3+2√2)4

81+2×4 +1/81+2×4

81+8+1/81+8

89+1/89

89×89+1/89

7922/89

89.01

Answer 2

Your Answer:-

$\tt Finding \ \ \dfrac{1}{x} \ \ first$

$\tt \dfrac{1}{x} = \dfrac{1}{ 3+2\sqrt2} \\\\ \tt Rationalizing \ \ the \ \ denominator \\\\ \tt \Rightarrow \dfrac{1}{3+2\sqrt2} \times \dfrac{3-2\sqrt2}{3-2\sqrt2} \\\\ \tt \Rightarrow \dfrac{3-2\sqrt2}{(3)^2-(2\sqrt2)} \\\\ \tt \Rightarrow \dfrac{3-2\sqrt2}{9-8} \\\\ \tt \Rightarrow 3-2\sqrt2$

$\tt (x + \dfrac{1}{x})^2 = (x)^2 + (\dfrac{1}{x})^2 + 2(x)(\dfrac{1}{x}) \\\\ \tt \Rightarrow (3+2\sqrt2 + 3 - 2\sqrt2)^2 = (x)^2 + (\dfrac{1}{x})^2 + 2 \\\\ \tt \Rightarrow 36 = (x)^2 + (\dfrac{1}{x})^2 + 2 \\\\ \tt \Rightarrow (x)^2 + (\dfrac{1}{x})^2 = 34$

Now Squaring both sides again.

$\tt (x^2 + \dfrac{1}{x^2})^2 = (x^2)^2 + (\dfrac{1}{x^2})^2 + 2(x^2)(\dfrac{1}{x^2}) \\\\ \tt \Rightarrow (x^2)^2 + (\dfrac{1}{x^2})^2 + 2(x^2)(\dfrac{1}{x^2}) = (34)^2 \\\\ \tt \Rightarrow x^4 + \dfrac{1}{x^4}+ 2 = (34)^2 \\\\ \tt \Rightarrow x^4 + \dfrac{1}{x^4}+ 2 = 1156 \\\\ \tt \Rightarrow x^4 + \dfrac{1}{x^4} = 1154$