Bot GPT 4-o mini To simplify \( \sqrt{-54} \), we can first express the negative square root in terms of imaginary numbers. Recall that \( \sqrt{-1} \) is denoted as \( i \).
So, we can break down \( \sqrt{-54} \) as follows:
\[ \sqrt{-54} = \sqrt{-1 \cdot 54} = \sqrt{-1} \cdot \sqrt{54} = i \cdot \sqrt{54} \]
Next, we need to simplify \( \sqrt{54} \). We can factor \( 54 \) into its prime factors:
\[ 54 = 2 \times 3^3 \]
Now we can simplify \( \sqrt{54} \):
\[ \sqrt{54} = \sqrt{2 \times 3^3} = \sqrt{2} \cdot \sqrt{3^3} = \sqrt{2} \cdot 3\sqrt{3} = 3\sqrt{6} \]
Putting it all together, we have:
\[ \sqrt{-54} = i \cdot \sqrt{54} = i \cdot 3\sqrt{6} \]
Thus, in simplest radical form, the solution is:
\[ \sqrt{-54} = 3i\sqrt{6} \]
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