Simplify the square root of -140.

Step-by-step solution

1. 1

   Extract the imaginary unit before anything else. By definition $i=\sqrt{-1}$, so for $a>0$:

   $\sqrt{-a}=i\sqrt{a}\quad\Longrightarrow\quad\sqrt{-140}=i\sqrt{140}$

   Doing this first keeps every later step inside the real numbers, where the usual radical rules are valid.

2. 2

   Find the largest perfect-square factor of $140$. Its prime factorisation is $140=2^2\times 5\times 7$, so the square part is $4$:

   $140=4\times 35$

3. 3

   Simplify the real radical.

   $\sqrt{140}=\sqrt4\cdot\sqrt{35}=2\sqrt{35}$

   and $35=5\times 7$ is square-free, so nothing more comes out.

4. 4

   Combine the pieces.

   $\sqrt{-140}=2i\sqrt{35}$

5. 5

   Check by squaring. $\left(2i\sqrt{35}\right)^2=4i^2\cdot 35=4(-1)(35)=-140$ $\checkmark$. Numerically $2\sqrt{35}\approx 11.8322$, so the value is about $11.8322i$.