simplify-t-26-3-3-5-1-3-26-3-3-5-1-3-

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}}]

\dots .

Answered by som(math1967) last updated on 30/Sep/21

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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