if-a-b-c-2196-a-1-3-b-c-2076-a-b-1-3-c-1860-a-b-c-1-3-480-determine-the-value-of-a-2-3-b-2-3-c-2-3-if-a-b-c-are-all-integer-

Question Number 171944 by infinityaction last updated on 22/Jun/22

i f

a + b + c = 2196

\sqrt[3]{a} + b + c = 2076

a + \sqrt[3]{b} + c = 1860

a + b + \sqrt[3]{c} = 480, d e t e r m i n e t h e v a l u e o f

a^{\frac{2}{3}} + b^{\frac{2}{3}} + c^{\frac{2}{3}}, i f a, b, c a r e a l l i n t e g e r .

Commented by infinityaction last updated on 22/Jun/22

r e p o s t o f q u e . n o . 171919

Commented by infinityaction last updated on 22/Jun/22

218

Commented by infinityaction last updated on 22/Jun/22

l e t^{3} \sqrt{a} = x,^{3} \sqrt{b} = y a n d^{3} \sqrt{c} = z

x^{3} + y^{3} + z^{3} = 2196 \dots \dots . (1)

x + y^{3} + z^{3} = 2076 \dots \dots \dots . (2)

x^{3} + y + z^{3} = 1860 \dots \dots \dots (3)

x^{3} + y^{3} + z = 480 \dots \dots . (4)

{e q}^{n} (1) - {e q}^{n} (2)

x^{3} - x = 120

(x - 1) x (x + 1) = 120

a, b, c \in Z t h e n x, y, z \in Z

(x - 1), x, (x + 1) a r e i n A . P

s o f a c t o r o f 120 = 4 \times 5 \times 6

s o x = 5

s i m i l a r l y

(y - 1) y (y + 1) = 6 \times 7 \times 8 = 336

y = 7

a n d

(z - 1) z (z + 1) = 11 \times 12 \times 13 = 1716

z = 12

^{3} \sqrt{a} = x \Rightarrow a^{2 / 3} = x^{2}

a^{2 / 3} = 25 \dots . (4)

s i m i l a r l y

b^{2 / 3} = 49 \dots . . (5)

a n d c^{2 / 3} = 144 \dots (6)

{e q}^{n} (4) + {e q}^{n} (5) + {e q}^{n} (6)

a^{2 / 3} + b^{2 / 3} + c^{2 / 3} = 25 + 49 + 144

a^{2 / 3} + b^{2 / 3} + c^{2 / 3} = 218

Commented by Tawa11 last updated on 22/Jun/22

Great sir

Commented by Mikenice last updated on 23/Jun/22

t h a n k s s i r