simplify t=(((26)/(3(√3)))+5)^(1/3) −(((26)/(3(√3)))−5)^(1/3)


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Question Number 155428 by ajfour last updated on 30/Sep/21

$s i m p l i f y$ $t = {(\frac{26}{3 \sqrt{3}} + 5)}^{1 / 3} - {(\frac{26}{3 \sqrt{3}} - 5)}^{1 / 3}$
	Answered by peter frank last updated on 30/Sep/21

	$t = {(\frac{26 + 15 \sqrt{3}}{3 \sqrt{3}})}^{\frac{1}{3}} - {(\frac{26 - 15 \sqrt{3}}{3 \sqrt{3}})}^{\frac{1}{3}}$ $t = \frac{{(26 + 15 \sqrt{3})}^{\frac{1}{3}}}{^{3} \sqrt{81}} - \frac{{(26 - 15 \sqrt{3})}^{\frac{1}{3}}}{^{3} \sqrt{81}}$ $t = \frac{1}{\sqrt{81}} [{(26 + 15 \sqrt{3})}^{\frac{1}{3}}] - [{(26 - 15 \sqrt{3})}^{\frac{1}{3}}]$ $. . . .$
	Answered by som(math1967) last updated on 30/Sep/21
	$let a=(((26)/(3(√3)))+5)^(1/3) b=(((26)/(3(√3)))−5)^(1/3) ∴a^3 −b^3 =((26)/(3(√3)))+5−((26)/(3(√3)))+5 a^3 −b^3 =10 ab={((26^2 )/(27))−25}^(1/3) =((1/(27)))^(1/3) =(1/3) (a−b)^3 =a^3 −b^3 −3ab(a−b) x^3 =10−3×(1/3)×x [let (a−b)=x] x^3 +x−10=0 x^3 −2x^2 +2x^2 −4x+5x−10=0 (x−2)(x^2 +2x+5)=0 x=2 [taking real value only] ∴a−b=2 ∴(((26)/(3(√3))) +5)^(1/3) +(((26)/(3(√3))) −5)^(1/3) =2$
	$l e t a = {(\frac{26}{3 \sqrt{3}} + 5)}^{\frac{1}{3}} b = {(\frac{26}{3 \sqrt{3}} - 5)}^{\frac{1}{3}}$ $∴ a^{3} - b^{3} = \frac{26}{3 \sqrt{3}} + 5 - \frac{26}{3 \sqrt{3}} + 5$ $a^{3} - b^{3} = 10$ $a b = {\frac{26^{2}}{27} - 25}^{\frac{1}{3}} = {(\frac{1}{27})}^{\frac{1}{3}} = \frac{1}{3}$ ${(a - b)}^{3} = a^{3} - b^{3} - 3 a b (a - b)$ $x^{3} = 10 - 3 \times \frac{1}{3} \times x [l e t (a - b) = x]$ $x^{3} + x - 10 = 0$ $x^{3} - 2 x^{2} + 2 x^{2} - 4 x + 5 x - 10 = 0$ $(x - 2) (x^{2} + 2 x + 5) = 0$ $x = 2 [t a k i n g r e a l v a l u e o n l y]$ $∴ a - b = 2$ $∴ {(\frac{26}{3 \sqrt{3}} + 5)}^{\frac{1}{3}} + {(\frac{26}{3 \sqrt{3}} - 5)}^{\frac{1}{3}} = 2$
	Commented by ajfour last updated on 30/Sep/21

	$t h a n k s s i r, a l l t o o n i c e .$
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