# If ${a^2} + {b^2} + {c^2} = ab + bc + ca,$ then find the value of ${a^3} + {b^3}+{c^3}.$

A. $3{(abc)^3}$

B. 3abc

C. $3{a^2}{b^2}{c^2}$

D. None of these

Last updated date: 19th Mar 2023

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Answer

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Hint: we are going to use the basic algebraic formula to solve the given question.

$$\because {a^3} + {b^3} + {c^3} - 3abc$$

$ = ({a^2} + {b^2} + {c^2} - ab - bc - ca)(a + b + c)$

$ = {a^2} + {b^2} + {c^2}$

$ = ab + bc + ca$ (given)

${a^3} + {b^3} + {c^3} - 3abc = 0 \times (a + b + c)$

$ \Rightarrow {a^3} + {b^3} + {c^3} - 3abc = 0$

$\therefore {a^3} + {b^3} + {c^3} = 3abc$

Note: In algebra, we use alphabets like (x, y, z, a, b, c...) to substitute numbers in the equation to get the desired solution. Numbers are definite and their values are known. While alphabets are used to represent unknown numbers.

$$\because {a^3} + {b^3} + {c^3} - 3abc$$

$ = ({a^2} + {b^2} + {c^2} - ab - bc - ca)(a + b + c)$

$ = {a^2} + {b^2} + {c^2}$

$ = ab + bc + ca$ (given)

${a^3} + {b^3} + {c^3} - 3abc = 0 \times (a + b + c)$

$ \Rightarrow {a^3} + {b^3} + {c^3} - 3abc = 0$

$\therefore {a^3} + {b^3} + {c^3} = 3abc$

Note: In algebra, we use alphabets like (x, y, z, a, b, c...) to substitute numbers in the equation to get the desired solution. Numbers are definite and their values are known. While alphabets are used to represent unknown numbers.

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