Question

Without actually calculating the cubes, find the values of ${{\left( -12 \right)}^{3}}+{{7}^{3}}+{{5}^{3}}$.

Answer 1

Hint:Assume the given numbers as $a,b\text{ and }c$ and use the algebraic identity or expression given as: ${{a}^{3}}+{{b}^{3}}+{{c}^{3}}=\left( a+b+c \right)\left( {{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ac \right)+3abc$ to simplify the problem and get the answer.

Complete step-by-step answer:
In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B, which may contain some variables, produce the same value for all of the variables within a certain range of validity. In other words, A = B is an identity if A and B define the same functions, and an identity is an equality between functions that are differently defined. For example: ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ and ${{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1$ are identities. There are many common identities like: algebraic identity, trigonometric identity, logarithmic identity, exponential identity, etc., but here we have to use algebraic identity.
An algebraic identity is an equality that holds for any values of its variables. For example, the identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ holds for all values of $a\text{ and }b$.
Now, we come to the question. Let us assume that, $-12=a,7=b\text{ and }5=c$. Then,
$\begin{aligned} & {{a}^{3}}+{{b}^{3}}+{{c}^{3}}=\left( a+b+c \right)\left( {{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ac \right)+3abc \\\ & \therefore {{\left( -12 \right)}^{3}}+{{7}^{3}}+{{5}^{3}}=\left( -12+7+5 \right)\left( {{(-12)}^{2}}+{{7}^{2}}+{{5}^{2}}-(-12)\times 7-7\times 5-(-12)\times 5 \right)+3\times \left( -12 \right)\times 7\times 5 \\\ & \text{ }=0\times \left( {{(-12)}^{2}}+{{7}^{2}}+{{5}^{2}}-(-12)\times 7-7\times 5-(-12)\times 5 \right)-1260 \\\ & \text{ }=-1260 \\\ \end{aligned}$
Hence, the value of the given expression is -1260.

Note: We have used algebraic identity to solve the above question because it helped us to easily simplify the given expression. If we would not have used the identity then cubing of numbers like 12 would have been time consuming and it is also possible that we would have made a mistake.