Question

Use property to evaluate: \[{\left( {38} \right)^3} + {( - 26)^3} + {( - 12)^3}\]
(A) 34568
(B) 35567
(C) 35566
(D) 35568

Answer 1

Hint: According to the given question, the given equation \[{\left( {38} \right)^3} + {( - 26)^3} + {( - 12)^3}\] that is in the form of \[{a^3} + {b^3} + {c^3}\] . So, we will use the algebraic identity that contains this all terms of cube which is \[{a^3} + {b^3} + {c^3} - 3abc = (a + b + c)({a^2} + {b^2} + {c^2} - ab - bc - ac)\]. Hence, we will solve for the right hand side as well as the left hand side to get the required answer.

Formula used:
Here, we will use the algebraic identity that is \[{a^3} + {b^3} + {c^3} - 3abc = (a + b + c)({a^2} + {b^2} + {c^2} - ab - bc - ac)\]

Complete step-by-step answer:
As, it is given \[{\left( {38} \right)^3} + {( - 26)^3} + {( - 12)^3}\] .
As, we know the identity of algebraic that is \[{a^3} + {b^3} + {c^3} - 3abc = (a + b + c)({a^2} + {b^2} + {c^2} - ab - bc - ac)\]
Here, in the given equation a = 38 , b = \[ - 26\] and c = \[ - 12\]
As, now we will calculate right hand side of the identity that is \[ = (a + b + c)({a^2} + {b^2} + {c^2} - ab - bc - ac)\]
On substituting the values of a, b and c we get,
 \[ = (38 - 26 - 12)\left( {{{\left( {38} \right)}^2} + {{\left( { - 26} \right)}^2} + {{\left( { - 12} \right)}^2} - \left( {38 \times - 26} \right) - \left( { - 26 \times - 12} \right) - \left( {38 \times - 12} \right)} \right)\]
After simplifying first bracket we get,
\[ = 0\left( {{{\left( {38} \right)}^2} + {{\left( { - 26} \right)}^2} + {{\left( { - 12} \right)}^2} - \left( {38 \times - 26} \right) - \left( { - 26 \times - 12} \right) - \left( {38 \times - 12} \right)} \right)\]
\[ = 0\]
So we get right hand side that is \[(a + b + c)({a^2} + {b^2} + {c^2} - ab - bc - ac) = 0\]
Hence, the identity becomes \[{a^3} + {b^3} + {c^3} - 3abc = 0\]
Taking 3abc on the right side we get,
\[{a^3} + {b^3} + {c^3} = 3abc\]
As, the given equation \[{\left( {38} \right)^3} + {( - 26)^3} + {( - 12)^3}\] in the form of left hand side of the identity that is \[{a^3} + {b^3} + {c^3}\] .
So, we will substitute the values of a, b and c in the right hand side of the identity that is 3abc.
After substituting we get,
\[ = 3 \times 38 \times - 26 \times - 12\]
After multiplying all the values and making negative-negative as positive sign we get,
\[ = 35568\]
Hence, \[{\left( {38} \right)^3} + {( - 26)^3} + {( - 12)^3}\] \[ = 35568\]
So, option (D) 35568 is correct.

Note: To solve these types of questions, we must remember the algebraic identities to solve the given equation. The equation can be in the form of \[{a^2} - {b^2}\] for which we have to use the identity that is \[{a^2} - {b^2} = (a - b)(a + b)\] . Similarly it can be in the form of \[{a^3} - {b^3}\] as well as \[{a^3} + {b^3}\] . Just follow the same procedure as shown in the above question and use the identity according to the given question.