Question

Suppose x = 13/ root(19 + 8root3), find the exact value of x^4 - 6x^3 - 2x^2 + 18x + 23/ x^2 - 8x + 15

Answer 1

Answer:

5

Step-by-step explanation:

$x = \frac{13}{\sqrt{19 +8\sqrt{3} } } \\\\Denominator = {\sqrt{19 +8\sqrt{3} } } \\\\= \sqrt{4^2 + \sqrt{3}^2 + 2*4*\sqrt{3} } \\= \sqrt{(4+\sqrt{3})^2 } \\= 4 + \sqrt{3}$

$x = \frac{13}{4+\sqrt{3\\} } \\(Rationalisation)\\\\ \frac{13}{4+ \sqrt{3} } *\frac{4-\sqrt{3} }{4-\sqrt{3} } \\\\\frac{13(4-\sqrt{3) } }{13}$

$x = 4 - \sqrt{3} \\\\\frac{x^4 - 6x^3 - 2x^2+18x+23}{x^2-8x+15} \\\\x^4 =( (4+\sqrt{3}) ^{2} )^2$

$x^4 =( 19 + 8\sqrt{3} )^2\\x^4 = 553 + 304\sqrt{3} \\\\x^3 = (4+\sqrt{3)}^3 \\x^3 = 64 + 3\sqrt{3} + 48\sqrt{3}+36$

$x^3 = 100 +51\sqrt{3} \\\\x^2 = (4+\sqrt{3})^2\\x^2 = 19 + 8\sqrt{3}$

(Substitute the values in Numerator)

$(553+304\sqrt{3}) -6(100+51\sqrt{3}) - 2(19+8\sqrt{3}) +18(4+\sqrt{3})+23\\553+304\sqrt{3}-600-306\sqrt{3} -38 - 16\sqrt{3} +72+18\sqrt{3}+23\\ 10\\Denominator = x^2 -8x +15\\(x-3)(x-5)$

$(4+\sqrt{3}-3)(4+\sqrt{3}-5)\\(\sqrt{3} + 1)(\sqrt{3} - 1)\\2\\\\Answer = \frac{10}{2} = 5$