Question

Decimal expansion of 8/3 is? ${\left( {a - b} \right)^3} + \left[ {{a^3} - {b^3}} \right] = ?$

Answer 1

Hint: In this question, we have two parts to be solved. First part is the determination of the decimal equivalent of the fraction 8/3 while the second part is the expansion of the arithmetic expression ${\left( {a - b} \right)^3} + \left[ {{a^3} - {b^3}} \right] = ?$.
For the first question, we will use the division method to determine the expansion of 8/3 in decimal form.
For the second question, we will use the arithmetic identity ${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)$ to expand the given expression.

Complete step-by-step answer:
Following the long division rule to expand the fraction 8/3 as:
\[
  3)8(11.66{\text{ }} \\
  {\text{ - }}\underline {\text{3}} \\
  {\text{ 5}} \\
  {\text{ - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{3} }} \\
  {\text{ 20}} \\
  {\text{ - }}\underline {{\text{18}}} \\
  {\text{ 020}} \\
  {\text{ - }}\underline {{\text{18}}} \\
  {\text{ 02}} \\
 \]
Here, a point indicates that the additional zero has been put in expanding the fraction. For one decimal point, we can use an infinite number of 0’s in the expansion but only one at a time. For each additional zero (consequent zero’s) we need to put 0 in the quotient as well.
Hence, $\dfrac{8}{3} = 11.66666....$ is the decimal expansion of the fraction 8/3. It can be seen that the decimal expansion is not terminating and is repeating in nature so, we can also write it as:
$\dfrac{8}{3} = 11.\overline 6 $.
Moreover, to approximate the result, we can also write the decimal equivalent as:
$\dfrac{8}{3} = 11.67$.

The given function is ${\left( {a - b} \right)^3} + \left[ {{a^3} - {b^3}} \right] = ? - - - - (i)$
Using the arithmetic identity ${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)$ in the equation (i) we get
$
  {\left( {a - b} \right)^3} + \left[ {{a^3} - {b^3}} \right] = {a^3} - {b^3} - 3ab\left( {a - b} \right) + \left[ {{a^3} - {b^3}} \right] \\
   = {a^3} - {b^3} + {a^3} - {b^3} - 3ab\left( {a - b} \right) \\
  = 2\left( {{a^3} - {b^3}} \right) - 3ab\left( {a - b} \right) \\
 $
Hence,
${\left( {a - b} \right)^3} + \left[ {{a^3} - {b^3}} \right] = 2\left( {{a^3} - {b^3}} \right) - 3ab\left( {a - b} \right)$.

Note: Students should be careful while expanding the arithmetic expression as it depends on the sign convention which is followed as
$
   - \times - = + \\
   - \times + = - \\
   + \times - = - \\
   + \times + = + \\
 $