Question

Write square root of, minus, 147, end square root
−147
​
in simplest radical form.

Answer 1

To express \(\sqrt{-147}\) in simplest radical form, we first recognize that the square root of a negative number involves imaginary numbers. We can start by factoring out the negative sign:

\[
\sqrt{-147} = \sqrt{-1 \cdot 147} = \sqrt{-1} \cdot \sqrt{147}
\]

The square root of \(-1\) is defined as \(i\) (the imaginary unit). Thus, we have:

\[
\sqrt{-147} = i \cdot \sqrt{147}
\]

Next, we simplify \(\sqrt{147}\). We can factor \(147\) into its prime factors. The prime factorization of \(147\) is:

\[
147 = 3 \cdot 49 = 3 \cdot 7^2
\]

This allows us to simplify the square root:

\[
\sqrt{147} = \sqrt{3 \cdot 7^2} = \sqrt{3} \cdot \sqrt{7^2} = \sqrt{3} \cdot 7 = 7\sqrt{3}
\]

Now substituting back into our expression for \(\sqrt{-147}\):

\[
\sqrt{-147} = i \cdot 7\sqrt{3} = 7\sqrt{3}i
\]

Thus, the simplest radical form of \(\sqrt{-147}\) is:

\[
\boxed{7\sqrt{3}i}
\]