Question

$\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} = 0$ then ${\left( {a + b + c} \right)^3}$
A) 27abc
B) 3abc
C) 9abc
D) abc

Answer 1

Hint: First, move the cube root of c to the other side of the equation. After that, cube both sides and apply the formula ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$. After that substitute the values in the formula. Then move the $\left( {a + b + c} \right)$ on one side and the remaining values on the other side. After that take a cube on both sides. Then the resultant value will be the answer for this problem.

Complete step-by-step answer:
Given: - $\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} = 0$
Move $\sqrt[3]{c}$ on the other sides,
$\sqrt[3]{a} + \sqrt[3]{b} = - \sqrt[3]{c}$ ….. (1)
Take cube on both sides,
\[ \Rightarrow {\left( {\sqrt[3]{a} + \sqrt[3]{b}} \right)^3} = {\left( { - \sqrt[3]{c}} \right)^3}\]
Expand the left side by the formula ${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$,
$ \Rightarrow {\left( {\sqrt[3]{a}} \right)^3} + {\left( {\sqrt[3]{b}} \right)^3} + 3\left( {\sqrt[3]{a}}\right)\left( {\sqrt[3]{b}} \right)\left( {\sqrt[3]{a} + \sqrt[3]{b}} \right) = {\left( { - \sqrt[3]{c}} \right)^3}$
Substitute the value from equation (1),
$ \Rightarrow a + b + 3\left( {\sqrt[3]{a}} \right)\left( {\sqrt[3]{b}} \right)\left( { - \sqrt[3]{c}} \right) = - c$
Take c on the left side and $3\left( {\sqrt[3]{a}} \right)\left( {\sqrt[3]{b}} \right)\left( { - \sqrt[3]{c}}
\right)$ on the right side,
\[ \Rightarrow a + b + c = 3\sqrt[3]{{abc}}\]
Take cube on both sides,
$ \Rightarrow {\left( {a + b + c} \right)^3} = {\left( {3\sqrt[3]{{abc}}} \right)^3}$
Open the value on the right side,
$ \Rightarrow {\left( {a + b + c} \right)^3} = 27abc$
So, the value of ${\left( {a + b + c} \right)^3}$ is 27abc.

Hence, option (A) is the correct answer.

Note: Please keep in mind the signs of the numbers, i.e. negative or positive numbers, while taking cube roots on both sides because a negative number always has a negative cube root and a negative cube. Here, the signs are both taken as positive hence this aspect is not considered here.
Algebra is a combination of both numerical and letters. Both represent the unknown quantity of the practical situation where a formula is applied to. When the numbers and letters come together with the factorials and matrices, formulas are formed. Algebraic formulas are put to use in particular situations for solving them. One of the most widely used formulas in math is the formula of ${\left( {a + b}
\right)^3}$ and ${\left( {a - b} \right)^3}$.
${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)$
${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)$