Question

√-75 in simplest radical form.

Answer 1

To simplify \(\sqrt{-75}\), we can separate the negative sign from the square root. We know that \(\sqrt{-1} = i\), where \(i\) is the imaginary unit.

Thus, we can rewrite \(\sqrt{-75}\) as:

\[
\sqrt{-75} = \sqrt{75} \cdot \sqrt{-1} = \sqrt{75} \cdot i
\]

Next, we simplify \(\sqrt{75}\). The number 75 can be factored into prime factors:

\[
75 = 25 \times 3 = 5^2 \times 3
\]

Taking the square root, we have:

\[
\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}
\]

Now, substituting this back into our expression for \(\sqrt{-75}\):

\[
\sqrt{-75} = 5\sqrt{3} \cdot i
\]

Therefore, the simplest radical form of \(\sqrt{-75}\) is:

\[
5\sqrt{3} i
\]