Divide $a + b$ by $a^{1 / 3} + b^{1 / 3}$

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Solution

Step 1:Multiply $a^{\frac{1}{3}} + b^{\frac{1}{3}}$ with $a^{\frac{2}{3}}$ , to get $a$ , Remainder $= a^{\frac{2}{3}} b^{\frac{1}{3}} + b$
Step 2: Multiply $a^{\frac{1}{3}} + b^{\frac{1}{3}}$ with $b^{\frac{2}{3}}$ , to get $a^{\frac{1}{3}} b^{\frac{2}{3}} + b$ , Remainder $= 0$

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1 + (x y) + (x y)^{2} + (x y)^{3} + . . . .

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(a^{\frac{1}{3}} + b^{\frac{1}{3}}) (a^{\frac{2}{3}} - a^{\frac{1}{3}} . b^{\frac{1}{2}} + b^{\frac{2}{3}}) = ?

a^{2} - b^{2}

a^{1 / 2} - b^{1 / 2}

a^{\frac{1}{3}} + b^{\frac{1}{3}} + c^{\frac{1}{3}} = 0

(a + b + c) = 27 a b c

x

y^{2} = 4 a x

x^{2} = 4 b y

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