Monkey Raptor V.2

The basic form looks like this: $$√{a ± 2√b}$$ a is $(x+y)$ and b is $(x·y)$ $$√{(x + y) ± 2√(xy)} = √x ± √y$$ Let's go to the methods and examples. PLUS $$√{a + 2√b}$$ $$√{(x + y) + 2√(xy)} = √x + √y$$ With $x > 0$ and $y > 0$ Example $√{7 + 2√12}$ We have $x + y = 7$ and $xy = 12$ Factor pairs of $12$: $1 × 12$ $2 × 6$ $3 × 4$ We pick the last factor pair: $3 × 4$, because $3 + 4 = 7$ We have $x = 3$ and $y = 4$ Therefore, $√{7 + 2√12}$ can be simplified to $√3 + √4$ ➡️ $√3 + 2$ In conclusion: $√{7 + 2√12} = √3 + 2$ ✅ MINUS $$√{a - 2√b}$$ $$√{(x + y) - 2√(xy)} = √x - √y$$ With $x > 0$, $y > 0$, and $x > y$ Example $√{15 - 2√56}$ We have $x + y = 15$ and $xy = 56$ Factor pairs of $56$: $1 × 56$ $2 × 28$ $4 × 14$ $7 × 8$ We pick the last factor pair: $7 × 8$, because $7 + 8 = 15$ But since $x > y$, then ...