Question

How do you simplify\[\dfrac{\left( \sqrt{2}+\sqrt{5} \right)}{\left( \sqrt{2}-\sqrt{5} \right)}\]?

Answer 1

Hint: In the given question, we have been asked to simplify an expression. In order to simplify the given question, first we need to multiply the numerator and denominator by the conjugate of the denominator. Later we expand the numerator and denominator by using distributive property and then we simplify the expression after expanding. Then we will combine the like terms and simplify further. In this way, we will get our required solution.

Complete step-by-step solution:
We have given that,
\[\Rightarrow \dfrac{\left( \sqrt{2}+\sqrt{5} \right)}{\left( \sqrt{2}-\sqrt{5} \right)}\]
Multiply the numerator and denominator by\[\dfrac{\left( \sqrt{2}+\sqrt{5} \right)}{\left( \sqrt{2}+\sqrt{5} \right)}\], we obtain
\[\Rightarrow \dfrac{\left( \sqrt{2}+\sqrt{5} \right)}{\left( \sqrt{2}-\sqrt{5} \right)}\times \dfrac{\left( \sqrt{2}+\sqrt{5} \right)}{\left( \sqrt{2}+\sqrt{5} \right)}\]
Expand the above expression by using distributive property, we get
Distributive property of multiplication states that \[\left( a\times b \right)\left( c\times d \right)=ac+ad+bc+bd\].
\[\Rightarrow \dfrac{\left( \sqrt{2}+\sqrt{5} \right)\left( \sqrt{2}+\sqrt{5} \right)}{\left( \sqrt{2}\times \sqrt{2} \right)+\left( \sqrt{2}\times \sqrt{5} \right)+\left( -\sqrt{5}\times \sqrt{2} \right)+\left( -\sqrt{5}\times \sqrt{5} \right)}\]
Simplifying the denominator and expanding the numerator in above expression, we get
\[\Rightarrow \dfrac{\left( \sqrt{2}\times \sqrt{2} \right)+\left( \sqrt{2}\times \sqrt{5} \right)+\left( \sqrt{5}\times \sqrt{2} \right)+\left( \sqrt{5}\times \sqrt{5} \right)}{2+\sqrt{10}-\sqrt{10}-5}\]
Simplifying the numerator and denominator in the above expression, we get
\[\Rightarrow \dfrac{2+\sqrt{10}+\sqrt{10}+5}{2+\sqrt{10}-\sqrt{10}-5}\]
Combining the like terms in the above expression, we get
\[\Rightarrow \dfrac{7+2\sqrt{10}}{-3}\]
Rewrite the above expression as,
\[\Rightarrow -\dfrac{7}{3}+\dfrac{2\sqrt{10}}{3}\]
Therefore,
\[\Rightarrow \dfrac{\left( \sqrt{2}+\sqrt{5} \right)}{\left( \sqrt{2}-\sqrt{5} \right)}=-\dfrac{7}{3}+\dfrac{2\sqrt{10}}{3}\]
Hence it is the required solution.

Note: While solving these types of problems, students should remember that they need to factorize the given expression. Students need to be very careful while factorization, they need to write all the terms and the signs associated with it very explicitly and very carefully to avoid making errors. For factorization of the given expression while expanding these types of expression we will need to use the distributive property of multiplication. We need to multiply the numerator and denominator by the conjugate of the denominator so that we will get a real number in the denominator.