Question

The value of ${{45}^{3}}-{{65}^{3}}-{{20}^{3}}$ is …….
A. 175500
B. -191500
C. 170000
D. -170000

Answer 1

Hint: For solving this question you should know about the general mathematical calculations. This problem can be solved easily by finding the values of the cubes of the given numbers and we can solve this by using the formula of the cubic variables as ${{a}^{3}}-{{b}^{3}}-{{c}^{3}}$.

Complete step by step answer:
According to our question it is asked of us to find the value of ${{45}^{3}}-{{65}^{3}}-{{20}^{3}}$. As we know that the cubic values of the digits are the values equal to the values which we get by multiplying itself three times. And in this problem, we will find the same values by some process and then we will subtract them as in the order which is given in the question. And thus we will get the answer. But this process is not very easy. Here, there are many chances of mistakes but if we solve this with the help of formula then there are not many chances that our question goes wrong and the formula for that is:
${{a}^{3}}-{{b}^{3}}-{{c}^{3}}={{\left( a-b-c \right)}^{3}}+3ab\left( a-b \right)+3bc\left( b+c \right)+3ac\left( a-c \right)-6ab$
And here the values of a, b and c are,
$a=45,b=65,c=20$
So, the values of ${{45}^{3}}-{{65}^{3}}-{{20}^{3}}$ is:
$\begin{align}
  & {{\left( -40 \right)}^{3}}+3\times 2925\left( -20 \right)+3\times 1300\left( 85 \right)+3\times 900\left( 25 \right)-6\times 58500 \\
 & =-191500 \\
\end{align}$

So, the correct answer is “Option B”.

Note: While solving this question there are two methods which can be used for solving the question and these are formula based and another is by solving the values. But the most useful and most accurate method is formula based because there is very less chance of being wrong.