Question

Find the value of x^4 +1÷x^4 if x=7+4√3

Answer 1

Answer:

x⁴ + 1/x⁴ = 37634

Step-by-step explanation:

Given :

• $\tt x = 7 + 4 \sqrt{3}$

$\tt \implies \frac{1}{x} = \frac{1}{7 + 4\sqrt{3} } \\ \\ \tt \implies \frac{1}{x} = \frac{1}{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } \\ \\ \tt \implies \frac{1}{x} = \frac{7 - 4 \sqrt{3} }{(7)^{2} - (4 \sqrt{3}) ^{2} } \\ \\ \tt \implies \frac{1}{x} = \frac{7 - 4 \sqrt{3} }{49 - 48} \\ \\ \tt \implies \frac{1}{x} = \frac{7 - 4 \sqrt{3} }{1} \\ \\ \tt \implies \frac{1}{x} = 7 - 4 \sqrt{3}$

Now, adding both :

$\tt x + \frac{1}{x} = 7 + 4 \sqrt{3} + 7 - 4 \sqrt{3} \\ \\ \tt \implies x + \frac{1}{x} = 7 + 7 \\ \\ \tt \implies x + \frac{1}{x} = 14$

On squaring both sides,

$\tt \implies (x + \frac{1}{x}) {}^{2} = (14) {}^{2} \\ \\ \tt \implies x {}^{2} + \frac{1}{x {}^{2} } + 2.x. \frac{1}{x} = 196 \\ \\ \tt \implies x {}^{2} + \frac{1}{x {}^{2} } + 2 = 196 \\ \\ \tt \implies x {}^{2} + \frac{1}{x {}^{2} } = 196 - 2 \\ \\ \tt \implies x {}^{2} + \frac{1}{x {}^{2} } = 194$

Again, on squaring both sides.

$\tt \implies ( x {}^{2} + \frac{1}{x {}^{2} }) {}^{2} = (194) {}^{2} \\ \\ \tt \implies x {}^{4} + \frac{1}{x {}^{4} } + 2.x {}^{2}. \frac{1}{x {}^{2} } = 37636 \\ \\ \tt \implies x {}^{4} + \frac{1}{x {}^{4} } + 2 = 37636\\ \\ \tt \implies x {}^{4} + \frac{1}{x {}^{4} } = 37636 - 2 \\ \\ \therefore \boxed{ \bf \red{ x {}^{4} + \frac{1}{x {}^{4} } = 37634}}$

Answer 2

HELLO MATE YOUR ANSWER IS⬇️

➡️x⁴ + 1/x⁴ = 37634⬅️