Simplify the following expression: \[6+\left\{ \dfrac{4}{3}+\left( \dfrac{3}{4}-\dfrac{1}{3} \right) \right\}\]

Hint: We will use the concept of LCM a few number of times to solve this question. We will begin from the extreme right and will proceed towards left after each step. Least common multiple (LCM)is the smallest positive integer that is divisible by both a and b.

Complete step-by-step answer:
The given expression is,
\[\Rightarrow 6+\left\{ \dfrac{4}{3}+\left( \dfrac{3}{4}-\dfrac{1}{3} \right) \right\}..........(1)\]
Now we will start with opening the brackets and then solving them.
Now taking the LCM of 4, 3 in equation (1) gives 12. And now dividing this LCM with denominator of respective terms and multiplying with the respective numerators what we get after dividing in equation (1). So using this information we get,
\[\Rightarrow 6+\left\{ \dfrac{4}{3}+\left( \dfrac{3\times 3-1\times 4}{12} \right) \right\}..........(2)\]
Now multiplying and subtracting in equation (2) we get,
\[\Rightarrow 6+\left\{ \dfrac{4}{3}+\left( \dfrac{9-4}{12} \right) \right\}=6+\left\{ \dfrac{4}{3}+\dfrac{5}{12} \right\}.......(3)\]
Now again taking the LCM of 3, 12 in equation (3) gives 12. And now again dividing this LCM with denominator of respective terms and multiplying with the respective numerators what we get after dividing in equation (3). So using this information we get,
\[\Rightarrow 6+\left\{ \dfrac{4\times 4+5\times 1}{12} \right\}.......(4)\]
Now multiplying and adding the terms in the numerator in equation (4) we get,
\[\Rightarrow 6+\left\{ \dfrac{16+5}{12} \right\}=6+\dfrac{21}{12}.......(5)\]
Now finally taking the LCM of 1, 12 in equation (5) gives 12. And now again dividing this LCM with denominator of respective terms and multiplying with the respective numerators what we get after dividing in equation (5). So using this information we get,
\[\Rightarrow \dfrac{12\times 6+21\times 1}{12}.......(6)\]
Now again multiplying and adding the terms and simplifying in equation (6) we get,
\[\Rightarrow \dfrac{72+21}{12}=\dfrac{93}{12}.......(7)\]
Now bringing it to the lowest simplest terms in equation (7) by dividing the numerator and denominator by 3, we get,
\[\Rightarrow \dfrac{93}{12}=\dfrac{31}{4}\]
Hence \[\dfrac{31}{4}\] is the answer.

Note: Understanding the concept of Least common multiple and the opening of brackets is the key here. We in a hurry can calculate the LCM wrongly in equation (1), (3) and (5) and hence we need to be careful while doing these steps.