Bot GPT 4-o mini 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:
\[ 5\sqrt{3} i \]
1 answer