Answer:
A+B+C+D = 13
Step-by-step explanation:
The given expression is:
[tex]\dfrac{1+\sqrt{2}}{2+\sqrt{3}}[/tex]
We have to simply it and express it in the form of:
[tex]A(1+\sqrt{B})-(\sqrt{C}+\sqrt{D})[/tex]
Multiply and divide the given expression with [tex]2-\sqrt 3[/tex]:
[tex]\dfrac{1+\sqrt{2}}{2+\sqrt{3}} \times \dfrac{2-\sqrt 3}{2-\sqrt 3}\\\Rightarrow \dfrac{(1+\sqrt{2}) \times (2-\sqrt 3)}{(2+\sqrt{3})\times (2-\sqrt 3)}\\\Rightarrow \dfrac{2+2\sqrt2-\sqrt3-\sqrt6}{2^2-(\sqrt{3})^2}\\\Rightarrow \dfrac{2+2\sqrt2-\sqrt3-\sqrt6}{4-3}\\\Rightarrow \dfrac{2(1+\sqrt2)-(\sqrt3+\sqrt6)}{1}\\\Rightarrow 2(1+\sqrt2)-(\sqrt3+\sqrt6)[/tex]
It is the simplified form of given expression.
Formula used:
[tex](a+b)(a-b) = a^{2} -b^{2}[/tex]
Comparing the simplified expression with [tex]A(1+\sqrt{B})-(\sqrt{C}+\sqrt{D})[/tex]
[tex]2(1+\sqrt2)-(\sqrt3+\sqrt6)=A(1+\sqrt{B})-(\sqrt{C}+\sqrt{D})\\\Rightarrow A =2, B=2, C=3\ and\ D=6[/tex]
So, value of
[tex]A+B+C+D = 2+2+3+6 = 13[/tex]
