Question

Find the value of the expression \[{\left( {0.2} \right)^3} - {\left( {0.3} \right)^3} + {\left( {0.1} \right)^2}\].

Answer 1

Hint: In these questions we will make use of the formula if \[a + b + c = 0\] then \[{a^3} + {b^3} + {c^3} = 3abc\], and here remember that \[ - {a^3}\] can be rewritten as \[{\left( { - a} \right)^3}\] i.e., \[ - {a^3} = {\left( { - a} \right)^3}\], and further simplify to get the required answer.

Complete step-by-step answer:
Given expression is \[{\left( {0.2} \right)^3} - {\left( {0.3} \right)^3} + {\left( {0.1} \right)^2}\],
Here as we know that \[ - {a^3}\] can be rewritten as \[{\left( { - a} \right)^3}\]i.e., \[ - {a^3} = {\left( { - a} \right)^3}\] so, here \[ - {\left( {0.3} \right)^3}\] can be rewritten as \[{\left( { - 0.3} \right)^3}\], so the given expression becomes,
$\Rightarrow$ \[{\left( {0.2} \right)^3} + {\left( { - 0.3} \right)^3} + {\left( {0.1} \right)^2}\]
Now using the formula if \[a + b + c = 0\] then \[{a^3} + {b^3} + {c^3} = 3abc\], now first we have to find the value of a+b+c , so here
$\Rightarrow$ \[a = 0.2\], \[b = - 0.3\] and \[c = 0.1\], substituting the values we get,
$\Rightarrow$ \[a + b + c = 0.2 - 0.3 + 0.1\],
Now simplifying we get,
\[a + b + c = 0\], so condition is satisfied we can now use the formula, if \[a + b + c = 0\] then \[{a^3} + {b^3} + {c^3} = 3abc\],
Now substituting the values in the second formula, we get
$\Rightarrow$ \[{a^3} + {b^3} + {c^3} = 3abc\],
Now substituting the values of a, b and c we get,
\[ \Rightarrow {\left( {0.2} \right)^3} + {\left( { - 0.3} \right)^3} + {\left( {0.1} \right)^3} = 3\left( {0.2} \right)\left( { - 0.3} \right)\left( {0.1} \right)\],
Now further multiplying we get,
\[ \Rightarrow {\left( {0.2} \right)^3} + {\left( { - 0.3} \right)^3} + {\left( {0.1} \right)^3} = 3\left( { - 0.06} \right)\left( {0.1} \right)\],
Now again multiplying we get,
\[ \Rightarrow {\left( {0.2} \right)^3} + {\left( { - 0.3} \right)^3} + {\left( {0.1} \right)^3} = 3\left( { - 0.006} \right)\],
Finally multiplying we get,
\[ \Rightarrow {\left( {0.2} \right)^3} + {\left( { - 0.3} \right)^3} + {\left( {0.1} \right)^3} = - 0.018\]

\[\therefore \]The value of \[{\left( {0.2} \right)^3} - {\left( {0.3} \right)^3} + {\left( {0.1} \right)^3} = - 0.018\].

Note:
These types of questions can also be solved by directly using cubes of the given numbers as it is not mentioned whether we should only use the formula or we should only use the cubes. So this can be solved in another method by evaluating cubes i.e.,
Given, \[{\left( {0.2} \right)^3} - {\left( {0.3} \right)^3} + {\left( {0.1} \right)^3}\],
Now evaluating the cubes of each term we get,
$\Rightarrow$ \[{\left( {0.2} \right)^3} - {\left( {0.3} \right)^3} + {\left( {0.1} \right)^3} = 0.008 - 0.027 + 0.001\]
So now simplifying R.H.S we get,
$\Rightarrow$\[{\left( {0.2} \right)^3} - {\left( {0.3} \right)^3} + {\left( {0.1} \right)^3} = 0.009 - 0.027\],
Again simplifying we get,
$\Rightarrow$\[{\left( {0.2} \right)^3} - {\left( {0.3} \right)^3} + {\left( {0.1} \right)^3} = - 0.018\].
So in both the cases we got the same answer, so it is up to the question what is asked and if it is not specified in the question it is up to the students which method to be used and what formula to be used to solve these types of questions.