Without calculating the cubes of the numbers, find the value of \[{{\left( -10 \right)}^{3}}+{{3}^{3}}+{{7}^{3}}\]

Hint: To solve the question, we have to apply the formula of sum of cubes of three numbers. Substitute the values in the formula to calculate the value of the given expression.

Complete step-by-step answer:
The given expression is \[{{\left( -10 \right)}^{3}}+{{3}^{3}}+{{7}^{3}}\]
We know the formula for sum of cubes of three numbers a, b, c is equal to
\[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}=\left( a+b+c \right)\left( {{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca \right)+3abc\] ….. (1)
On comparing with the given numbers, we get
a = -10, b = 3, c = 7
By substituting the above values in the \[\left( a+b+c \right)\]we get
\[\begin{align}
  & a+b+c=-10+7+3 \\
 & =-10+10 \\
 & =0 \\
\end{align}\]
Thus, we get the value of \[a+b+c\] is equal to 0
By substituting the above values in the \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca\] we get
\[\begin{align}
  & {{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca \\
 & ={{(-10)}^{2}}+{{3}^{2}}+{{7}^{2}}-(-10\times 3)-(3\times 7)-(-10\times 7) \\
 & =100+9+49-(-30)-21-(-70) \\
 & =100+58+30-21+70 \\
 & =188+79 \\
 & =267 \\
\end{align}\]
Thus, we get the value of \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca\] is equal to 297.
By substituting the above values in the \[3abc\] we get
\[\begin{align}
  & 3abc=3\times \left( -10 \right)\times 3\times 7 \\
 & =\left( -30 \right)\times 21 \\
 & =-630 \\
\end{align}\]
Thus, we get the value of \[3abc\] is equal to -630.
By substituting the above values in the formula in equation (1) we get
\[\begin{align}
  & {{\left( -10 \right)}^{3}}+{{3}^{3}}+{{7}^{3}}=\left( 0 \right)\left( 297 \right)-630 \\
 & =0-630 \\
 & =-630 \\
\end{align}\]
Thus, we get the value of \[{{\left( -10 \right)}^{3}}+{{3}^{3}}+{{7}^{3}}\] is equal to -630.

Note: The possibility of mistake can be not applying the formula of sum of cubes of three numbers, which is an important step since solving through calculating the cube values is not allowed. The other possibility of mistake can be the calculation mistake since the procedure of solving involves more calculations. This mistake can be avoided by rechecking the answer by calculating the cubes of the given numbers.