Question

Which expression is equivalent to (first picture is the equation the rest are the answers)

Answer 1

The prime factorization of 108 is

`108 = 2^2\cdot 3^3`

This means that we have

`√(-108) = √((-1)\cdot2^2\cdot 3^3) = √(-1)\cdot√(2^2)\cdot√(3^3) = √(-1)\cdot√(2^2)\cdot√(3^2\cdot 3) = √(-1)\cdot√(2^2)\cdot√(3^2)\cdot√(3)`

We can simplify the squares and the square roots:

`√(-1)\cdot√(2^2)\cdot√(3^2)\cdot√(3) = i\cdot 2\cdot 3 \cdot √(3) = 6i√(3)`

Similarly, we have

`√(-3) = √((-1)\cdot 3) = √(-1)\cdot√(3)=i√(3)`

So, the difference is

`√(-108)-√(-3) = 6i√(3)-i√(3) = 5i√(3)`