Question

Evaluate the given expression:
${{\left( \dfrac{1}{2} \right)}^{3}}+{{\left( \dfrac{1}{3} \right)}^{3}}-{{\left( \dfrac{5}{6} \right)}^{3}}$
A.$\dfrac{-5}{14}$
B.$\dfrac{-5}{12}$
C.$\dfrac{-7}{12}$
D.$\dfrac{-5}{2}$

Answer 1

Hint: We are given the expression, ${{\left( \dfrac{1}{2} \right)}^{3}}+{{\left( \dfrac{1}{3} \right)}^{3}}-{{\left( \dfrac{5}{6} \right)}^{3}}$. So, we will take the cubes of the terms separately and then solve it. After that, when we simplify it we will get the answer.

Complete step-by-step answer:
Now we are given in the question to evaluate the following expression, ${{\left( \dfrac{1}{2} \right)}^{3}}+{{\left( \dfrac{1}{3} \right)}^{3}}-{{\left( \dfrac{5}{6} \right)}^{3}}$.
We can solve the above expression, by first considering each terms one by one and then finding its cube and then by applying the respective operations, we will arrive at our final answer.
So on solving the expression, we get,
$\begin{align}
  & {{\left( \dfrac{1}{2} \right)}^{3}}+{{\left( \dfrac{1}{3} \right)}^{3}}-{{\left( \dfrac{5}{6} \right)}^{3}} \\
 & =\left( \dfrac{1}{8} \right)+\left( \dfrac{1}{27} \right)-\left( \dfrac{125}{216} \right) \\
\end{align}$
Now, we can perform the addition of the first two terms. So, for that, we will take the LCM of those two terms and find the value. Hence, we will get,
$=\left( \dfrac{27+8}{27\times 8} \right)-\left( \dfrac{125}{216} \right)$
On simplification, we will get,
$=\left( \dfrac{35}{216} \right)-\left( \dfrac{125}{216} \right)$
Now we will subtract both the terms. So, we will get,
$=\left( \dfrac{35-125}{216} \right)$
On simplification, we will get,
$=\left( \dfrac{-90}{216} \right)$
Now, by reducing the above term to its simplest form, we will get as follow,
$=\left( \dfrac{-5}{12} \right)$
Therefore, we get the answer of the expression, ${{\left( \dfrac{1}{2} \right)}^{3}}+{{\left( \dfrac{1}{3} \right)}^{3}}-{{\left( \dfrac{5}{6} \right)}^{3}}$ as $\dfrac{-5}{12}$.
Therefore the correct answer is option (B).

Note: The students must read the given question properly and see what is asked in the question. They must have a clear idea on the concepts regarding addition, division, multiplication and subtraction. A proper assumption should be made. The possible mistake that the students can do in this question, is making calculation mistakes. So, the students must be careful while performing the calculations.