Question

Simplify :
\[\dfrac{{\sqrt {18} }}{{5\sqrt {18} + 3\sqrt {72} + 2\sqrt {162} }}\]

Answer 1

Hint: Square root of a number is a value , which on multiplication by itself results out the original number . In the given question , try to solve the denominator part first . Now , if we look at the denominator we can simplify it by making the square root terms in the form of numerator i.e. in terms of \[\sqrt {18} \] and then solve it accordingly to get the required answer . If a number is a perfect square number, then there exists a perfect square root , the same we have to use in denominator terms.

Complete step by step answer:
Given : \[\dfrac{{\sqrt {18} }}{{5\sqrt {18} + 3\sqrt {72} + 2\sqrt {162} }}\]
Now simplifying the square root terms in denominator in terms of \[\sqrt {18} \] we get ,
\[ = \dfrac{{\sqrt {18} }}{{5\sqrt {18} + 3\sqrt {4 \times 18} + 2\sqrt {9 \times 18} }}\]
Now taking out the perfect square in denominator we get ,
\[ = \dfrac{{\sqrt {18} }}{{5\sqrt {18} + 3 \times 2\sqrt {18} + 2 \times 3\sqrt {18} }}\]
On simplifying the above expression we get ,
\[ = \dfrac{{\sqrt {18} }}{{5\sqrt {18} + 6\sqrt {18} + 6\sqrt {18} }}\]
Now taking \[\sqrt {18} \] common from the denominator we get ,
\[ = \dfrac{{\sqrt {18} }}{{\sqrt {18} \left( {5 + 6 + 6} \right)}}\]
On solving we get ,
\[ = \dfrac{{\sqrt {18} }}{{\sqrt {18} \left( {17} \right)}}\]
Now cancelling out the term \[\sqrt {18} \] from denominator and numerator we get ,
\[ = \dfrac{1}{{17}}\] .
Therefore , the simplification of the given expression is \[\dfrac{1}{{17}}\] .

Note:
Never try to solve the questions using the values of square root terms as it will produce inaccuracy in the solution. The questions in which the squares are present in denominator as well as in numerator , such types of questions are solved by obtaining common terms from numerator and denominator , and then cancelling them out . Try to make square root terms common so that they can be cancelled out and make the expression less complex . These questions are short and tricky .