To find: the simplified form of the given expression.
Solution:
We shall find the simplified form of the given expression in the following way.
We will use the method of rationalization of radical expression to find the simplified form.
Let us suppose we have a radical expression [tex]\frac{a}{\sqrt{b}-\sqrt{c} }[/tex]
We shall rationalize it to get its simplified form.
(Rationalization of a radical expression involves the multiplication with its conjugate keeping the value of the original expression unchanged.)
(The conjugate of any radical expression is obtained by interchanging the positive or negative signs before the radical part or the root part only and keeping the non-radical part absolutely the same.)
Therefore, we get the conjugate of [tex]\sqrt{b} -\sqrt{c}[/tex] as [tex]\sqrt{b} +\sqrt{c}[/tex].
We can rationalize the expression [tex]\frac{a}{\sqrt{b}-\sqrt{c} }[/tex] by multiplying its conjugate [tex]\sqrt{b}+\sqrt{c}[/tex] in the following way.
The required simplified form of [tex]\frac{6}{2\sqrt{3}-\sqrt{6}} +\frac{\sqrt{6} }{\sqrt{3}+\sqrt{2}} -\frac{4\sqrt{3} }{\sqrt{6} -\sqrt{2} } =0[/tex] which is the the final answer.
Answers & Comments
Verified answer
Given:
[tex]\frac{6}{2\sqrt{3}-\sqrt{6}} +\frac{\sqrt{6} }{\sqrt{3}+\sqrt{2}} -\frac{4\sqrt{3} }{\sqrt{6} -\sqrt{2} }[/tex]
To find: the simplified form of the given expression.
Solution:
We shall find the simplified form of the given expression in the following way.
We will use the method of rationalization of radical expression to find the simplified form.
Let us suppose we have a radical expression [tex]\frac{a}{\sqrt{b}-\sqrt{c} }[/tex]
We shall rationalize it to get its simplified form.
(Rationalization of a radical expression involves the multiplication with its conjugate keeping the value of the original expression unchanged.)
(The conjugate of any radical expression is obtained by interchanging the positive or negative signs before the radical part or the root part only and keeping the non-radical part absolutely the same.)
Therefore, we get the conjugate of [tex]\sqrt{b} -\sqrt{c}[/tex] as [tex]\sqrt{b} +\sqrt{c}[/tex].
We can rationalize the expression [tex]\frac{a}{\sqrt{b}-\sqrt{c} }[/tex] by multiplying its conjugate [tex]\sqrt{b}+\sqrt{c}[/tex] in the following way.
[tex]\frac{a}{\sqrt{b}-\sqrt{c} } \\=\frac{a\((\sqrt{b}+\sqrt{c})}{(\sqrt{b}-\sqrt{c})\((\sqrt{b}+\sqrt{c}) }\\=\frac{a\((\sqrt{b}+\sqrt{c})}{({b-c}) }[/tex]
We shall use this rationalization method to carry out the simplification.
[tex]\frac{6}{2\sqrt{3}-\sqrt{6}} +\frac{\sqrt{6} }{\sqrt{3}+\sqrt{2}} -\frac{4\sqrt{3} }{\sqrt{6} -\sqrt{2} }\\=\frac{6(2\sqrt{3}+\sqrt{6})}{(2\sqrt{3}-\sqrt{6})(2\sqrt{3}+\sqrt{6})} +\frac{\sqrt{6}(\sqrt{3}-\sqrt{2}) }{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})} -\frac{4\sqrt{3}(\sqrt{6} +\sqrt{2} ) }{(\sqrt{6} -\sqrt{2})(\sqrt{6} +\sqrt{2} ) }\\=\frac{6(2\sqrt{3}+\sqrt{6})}{4.3-6}} +\frac{\sqrt{6}(\sqrt{3}-\sqrt{2}) }{3-2} -\frac{4\sqrt{3}(\sqrt{6} +\sqrt{2} ) }{6-2} }\\[/tex]
We shall further simplify
[tex]=\frac{6(2\sqrt{3}+\sqrt{6})}{4.3-6}} +\frac{\sqrt{6}(\sqrt{3}-\sqrt{2}) }{3-2} -\frac{4\sqrt{3}(\sqrt{6} +\sqrt{2} ) }{6-2} }\\=(2\sqrt{3}+\sqrt{6})+\sqrt{6}(\sqrt{3}-\sqrt{2})-\sqrt{3}(\sqrt{6} +\sqrt{2} )\\=2\sqrt{3}+\sqrt{6}+\sqrt{18}-\sqrt{12}-\sqrt{18} -\sqrt{6}\\=2\sqrt{3}+\sqrt{6}+3\sqrt{3}-2\sqrt{3}-3\sqrt{3} -\sqrt{6}[/tex]
We find that ultimately all the terms cancel out and we get
[tex]\frac{6}{2\sqrt{3}-\sqrt{6}} +\frac{\sqrt{6} }{\sqrt{3}+\sqrt{2}} -\frac{4\sqrt{3} }{\sqrt{6} -\sqrt{2} }=0[/tex]
The required simplified form of [tex]\frac{6}{2\sqrt{3}-\sqrt{6}} +\frac{\sqrt{6} }{\sqrt{3}+\sqrt{2}} -\frac{4\sqrt{3} }{\sqrt{6} -\sqrt{2} } =0[/tex] which is the the final answer.