caigis72da

2023-03-17

What is the formula of $\left({a}^{3}+{b}^{3}\right)$?

uwafundebz3v

Beginner2023-03-18Added 8 answers

Find the formula for $\left({a}^{3}+{b}^{3}\right)$.

Since, for any real number $a\mathrm{and}b$

${\left(}^{a+b}={a}^{3}+{b}^{3}+3ab\left(a+b\right)\dots \left(1\right)$

Based on the equation$\left(1\right)$, the value ${a}^{3}+{b}^{3}$ the following is derived:

$\left({a}^{3}+{b}^{3}\right)={\left(a+b\right)}^{3}-3ab\left(a+b\right)\Rightarrow \left({a}^{3}+{b}^{3}\right)=\left(a+b\right)\left({\left(a+b\right)}^{2}-3ab\right)\dots \left(2\right)$

Replace with the value of ${\left(a+b\right)}^{2}$ from ${\left(}^{a+b}={a}^{2}+{b}^{2}+2ab$ in $\left(2\right)$:

$\left({a}^{3}+{b}^{3}\right)=\left(a+b\right)\left({a}^{2}+{b}^{2}+2ab-3ab\right)\Rightarrow \left({a}^{3}+{b}^{3}\right)=\left(a+b\right)\left({a}^{2}+{b}^{2}-ab\right)$

Thus, the formula for $\left({a}^{3}+{b}^{3}\right)$ is $\left({a}^{3}+{b}^{3}\right)=\left(a+b\right)\left({a}^{2}+{b}^{2}-ab\right)$.

Since, for any real number $a\mathrm{and}b$

${\left(}^{a+b}={a}^{3}+{b}^{3}+3ab\left(a+b\right)\dots \left(1\right)$

Based on the equation$\left(1\right)$, the value ${a}^{3}+{b}^{3}$ the following is derived:

$\left({a}^{3}+{b}^{3}\right)={\left(a+b\right)}^{3}-3ab\left(a+b\right)\Rightarrow \left({a}^{3}+{b}^{3}\right)=\left(a+b\right)\left({\left(a+b\right)}^{2}-3ab\right)\dots \left(2\right)$

Replace with the value of ${\left(a+b\right)}^{2}$ from ${\left(}^{a+b}={a}^{2}+{b}^{2}+2ab$ in $\left(2\right)$:

$\left({a}^{3}+{b}^{3}\right)=\left(a+b\right)\left({a}^{2}+{b}^{2}+2ab-3ab\right)\Rightarrow \left({a}^{3}+{b}^{3}\right)=\left(a+b\right)\left({a}^{2}+{b}^{2}-ab\right)$

Thus, the formula for $\left({a}^{3}+{b}^{3}\right)$ is $\left({a}^{3}+{b}^{3}\right)=\left(a+b\right)\left({a}^{2}+{b}^{2}-ab\right)$.

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