a^(1/3) +b^(1/3) +c^(1/3) =(1/(3^(1/3) −2^(1/3) )) b+c−a=?


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Question Number 217643 by Rojarani last updated on 17/Mar/25

$a^{\frac{1}{3}} + b^{\frac{1}{3}} + c^{\frac{1}{3}} = \frac{1}{3^{\frac{1}{3}} - 2^{\frac{1}{3}}}$ $b + c - a = ?$
	Answered by mehdee7396 last updated on 17/Mar/25

	$\frac{1}{\sqrt[3]{3} - \sqrt[3]{2}} = \sqrt[3]{9} + \sqrt[3]{6} + \sqrt[3]{4} \Rightarrow$ $a = 9 & b = 6 & c = 4 \Rightarrow b + c - a = 1$ $o r a = 6 & b = 9 & c = 4 u \Rightarrow b + c - a = 7$ $o r a = 4 & b = 9 & c = 6 \Rightarrow b + c - a = 11$ $1 o r 7 o r 11$
	Commented by Rojarani last updated on 17/Mar/25

	$S i r t h a n k s .$
	Answered by Rasheed.Sindhi last updated on 17/Mar/25

	$a^{\frac{1}{3}} + b^{\frac{1}{3}} + c^{\frac{1}{3}} = \frac{1}{3^{\frac{1}{3}} - 2^{\frac{1}{3}}}$ $b + c - a = ?$ $a^{\frac{1}{3}} + b^{\frac{1}{3}} + c^{\frac{1}{3}}$ $= \frac{1}{3^{\frac{1}{3}} - 2^{\frac{1}{3}}} \cdot \frac{3^{\frac{2}{3}} + 3^{\frac{1}{3}} \cdot 2^{\frac{1}{3}} + 2^{\frac{2}{3}}}{3^{\frac{2}{3}} + 3^{\frac{1}{3}} \cdot 2^{\frac{1}{3}} + 2^{\frac{2}{3}}}$ $= \frac{9^{\frac{1}{3}} + 6^{\frac{1}{3}} + 4^{\frac{1}{3}}}{{(3^{\frac{1}{3}})}^{3} - {(2^{\frac{1}{3}})}^{3}} = \frac{9^{\frac{1}{3}} + 6^{\frac{1}{3}} + 4^{\frac{1}{3}}}{3 - 2}$ $= 9^{\frac{1}{3}} + 6^{\frac{1}{3}} + 4^{\frac{1}{3}}$ $C a s e a = 9$ $b + c - a = 6 + 4 - 9 = 1 ✓$ $C a s e a = 6$ $b + c - a = 9 + 4 - 6 = 7 ✓$ $C a s e a = 4$ $b + c - a = 9 + 6 - 4 = 11 ✓$ $\begin{matrix} \overset{F o r m u l a u s e d}{(A - B) (A^{2} + A B + B^{2}) = A^{3} - B^{3}} \end{matrix}$
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