Question

Simplify $\frac{1+\sqrt{2}}{2+\sqrt{3}}$. Your solution can be converted to the form $A(1+\sqrt{B})-(\sqrt{C}+\sqrt{D})$, where $A$, $B$, $C$, and $D$ are positive integers. What is $A+B+C+D$?

Answer 1

A+B+C+D = 13

Explanation:

The given expression is:

`(1+√(2))/(2+√(3))`

We have to simply it and express it in the form of:

`A(1+√(B))-(√(C)+√(D))`

Multiply and divide the given expression with
`2-\sqrt 3`:

`(1+√(2))/(2+√(3)) * (2-\sqrt 3)/(2-\sqrt 3)\\\Rightarrow ((1+√(2)) * (2-\sqrt 3))/((2+√(3))* (2-\sqrt 3))\\\Rightarrow (2+2\sqrt2-\sqrt3-\sqrt6)/(2^2-(√(3))^2)\\\Rightarrow (2+2\sqrt2-\sqrt3-\sqrt6)/(4-3)\\\Rightarrow (2(1+\sqrt2)-(\sqrt3+\sqrt6))/(1)\\\Rightarrow 2(1+\sqrt2)-(\sqrt3+\sqrt6)`

It is the simplified form of given expression.

Formula used:

`(a+b)(a-b) = a^(2) -b^(2)`

Comparing the simplified expression with
`A(1+√(B))-(√(C)+√(D))`

`2(1+\sqrt2)-(\sqrt3+\sqrt6)=A(1+√(B))-(√(C)+√(D))\\\Rightarrow A =2, B=2, C=3\ and\ D=6`

So, value of

`A+B+C+D = 2+2+3+6 = 13`