Question

How do you simplify $\dfrac{{12 + \sqrt {108} }}{6}$?

Answer 1

Hint: Here we can convert the digit under the root in the simplest form and then we can cancel the common terms that will appear in the numerator and the denominator. Hence we will get the full term in the simplest form.

Complete step by step solution:
Here we are given to simplify the term which is given as $\dfrac{{12 + \sqrt {108} }}{6}$.
Here we need to convert the digit under the square root in the simplest form and then we need to cancel all the common terms from the numerator and the denominator and we will get the simplest form that is required.
So we can write $108$ as:
$
  108 = \left( 2 \right)\left( {54} \right) \\
  54 = \left( 2 \right)\left( {27} \right) \\
  27 = \left( 3 \right)\left( 9 \right) \\
  9 = \left( 3 \right)\left( 3 \right) \\
  3 = \left( 3 \right)\left( 1 \right) \\
 $
Hence we can say that $108 = \left( 2 \right)\left( 2 \right)\left( 3 \right)\left( 3 \right)\left( 3 \right)$
Now we can root as $\sqrt {108} = \sqrt {\left( 2 \right)\left( 2 \right)\left( 3 \right)\left( 3 \right)\left( 3 \right)} = \left( 2 \right)\left( 3 \right)\sqrt 3 = 6\sqrt 3 $
Now we can substitute this value we have calculated of the square root in the given fraction which we need to solve, we will get:
$\dfrac{{12 + \sqrt {108} }}{6}$
$\dfrac{{12 + 6\sqrt 3 }}{6}$
Now we can see that $6$ is common in the above fraction’s numerator and denominator as we can write $12 = \left( 6 \right)\left( 2 \right)$ and we will get:
$\dfrac{{\left( 2 \right)\left( 6 \right) + 6\sqrt 3 }}{6}$
Now cancelling the common terms from both the numerator and denominator of the above fraction we have got after the solving, we will get the simplified form as:
$\dfrac{{\left( 2 \right)\left( 6 \right) + 6\sqrt 3 }}{6} = 2 + \sqrt 3 $

So we get the simplified form of the above fraction as $2 + \sqrt 3 $

Note:
Whenever we have the problem to simplify the fraction which contains a square root, we just need to convert the square root portion in as much simplified form as we can and then just cancel the common terms from the whole fraction.