Bot GPT 4-o mini To simplify the expression \(\sqrt{-50}\), we can use the property of imaginary numbers.
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First, we can break \(-50\) into its factors: \[ -50 = -1 \times 50 \]
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We know that \(\sqrt{-1} = i\), where \(i\) is the imaginary unit. Thus, we can write: \[ \sqrt{-50} = \sqrt{-1 \times 50} = \sqrt{-1} \times \sqrt{50} = i \sqrt{50} \]
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Next, we simplify \(\sqrt{50}\). The number \(50\) can be factored as: \[ 50 = 25 \times 2 = 5^2 \times 2 \]
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Therefore: \[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} \]
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Substituting this back into the expression gives: \[ \sqrt{-50} = i \sqrt{50} = i \times 5\sqrt{2} = 5i\sqrt{2} \]
Thus, the simplest radical form of \(\sqrt{-50}\) is: \[ \boxed{5i\sqrt{2}} \]
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