Evaluate `sqrt(4+3sqrt(-20))+sqrt(4-3sqrt(-20))`.

1.	Evaluate `sqrt(4+3sqrt(-20))+sqrt(4-3sqrt(-20))`.
Answer» `sqrt(4+3sqrt(-20))=sqrt(4+6 sqrt(5)i)and sqrt(4-3 sqrt(-20))=sqrt(4-6sqrt(5)i)`. Let `sqrt(4+6 sqrt(5)i)=(x+iy)." "...(i)` On squaring both sides of (i), we get `(4+6 sqrt(5)i)=(x+iy)^(2)rArr (4+6 sqrt(5)i)=(x^(2)-y^(2))+i(2xy)." "...(ii)` On comparing real parts and imaginary parts on both sides of (ii), we get `(x^(2)-y^(2))=4 and 2xy=6 sqrt(5)`. `therefore" "(x^(2)+y^(2))=sqrt((x^(2)-y^(2))^(2)+4x^(2)y^(2))=sqrt((4)^(2)+(6 sqrt(5))^(2))=sqrt(16+180)=sqrt(196)=14`. Thus, `(x^(2)-y^(2))=4" "...(iii) and (x^(2)+y^(2))=14" "...(iv)`. On solving (iii) and (iv), we get `x^(2)=9 and y^(2)=5`. `therefore" "x = +- 3 and y = +- sqrt(5)`. Since `xy gt 0`, so x and y are of the same sign. `therefore" "(x = 3, y = sqrt(5))or(x = -3, y = -sqrt(5))`. `therefore" "sqrt(4+3 sqrt(-20))=sqrt(4+6 sqrt(5)i)=(3-sqrt(5)i)or(-3+sqrt(5)i)`. Similarly, `sqrt(4-3 sqrt(-20))=sqrt(4-6 sqrt(5)i)=(3-sqrt(5)i)or(-3+sqrt(5)i)`. `therefore" "sqrt(4+3 sqrt(-20))+sqrt(4-3 sqrt(-20))={{:({(3+sqrt(5)i)+(-3+sqrt(5)i)}=6),(" "or),({-3-sqrt(5)i}+{-3+sqrt(5)i}=-6):}` Hence, `sqrt(4+3sqrt(-20))+sqrt(4-3 sqrt(-20))=6 or -6`.

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