Using suitable identity, evaluate\[{\left( { - 32} \right)^3} + {\left( {18} \right)^3} + {\left( {14} \right)^3}\]

Hint: This question involves the arithmetic operations like addition/ subtraction/ multiplication/ division. We need to know the basic algebraic formulae to solve this type of question. Also, we need to know how to perform arithmetic operations with the involvement of different signs to make an easy calculation.

Complete step by step solution:
The given equation is shown below,
\[{\left( { - 32} \right)^3} + {\left( {18} \right)^3} + {\left( {14} \right)^3} = ? \to \left( 1 \right)\]
Let’s take, \[a = - 32,b = 18,c = 14\]
Let’s find\[a + b + c\]by using the above values. So, we get
\[a + b + c = - 32 + 18 + 14\]
By solving the above equation we get,
\[a + b + c = 0 \to \left( 2 \right)\]
We know that,
If\[a + b + c = 0\],
 Then
\[{a^3} + {b^3} + {c^3} = 3abc \to \left( 3 \right)\]
By comparing the equation\[\left( 1 \right)\]and\[\left( 3 \right)\], we get
\[\left( 1 \right) \to {\left( { - 32} \right)^3} + {\left( {18} \right)^3} + {\left( {14} \right)^3} = ?\]
\[\left( 3 \right) \to {a^3} + {b^3} + {c^3} = 3abc\]
So, we get\[a = - 32,b = 18,c = 14\]
Let’s substitute these values in the equation\[\left( 3 \right)\], we get
\[\left( 3 \right) \to {a^3} + {b^3} + {c^3} = 3abc\]
\[{\left( { - 32} \right)^3} + {\left( {18} \right)^3} + {\left( {14} \right)^3} = 3 \times \left( { - 32} \right) \times \left( {18} \right) \times \left( {14} \right)\]
By solving the above equation, we get
\[{\left( { - 32} \right)^3} + {\left( {18} \right)^3} + {\left( {14} \right)^3} = - 24192\]
So, the final answer is,
\[{\left( { - 32} \right)^3} + {\left( {18} \right)^3} + {\left( {14} \right)^3}\]is equal to\[ - 24192\]

Note: This question describes the arithmetic operations like addition/ subtraction/ multiplication/ division. Note that if\[a + b + c = 0\], then\[{a^3} + {b^3} + {c^3} = 3abc\]. To solve this type of question we would compare the given expression with the related algebraic formula to make the easy calculation. Note that anything power zero will be one and zero power anything will be zero. Also, note the following things when multiply or divide different sign terms,
1. When a negative term is multiplied or divided by the negative term the final answer will be a positive term.
2. When a positive term is multiplied or divided by the negative term the final answer will be a negative term.
3. When a positive term is multiplied or divided by the positive term the final answer will be a positive term.