Simplify the given square root. $\sqrt{-75}$

• A. $25i\sqrt3$
• B. $5i\sqrt3$
• C. $3i\sqrt5$
• D. $75i$

Answer: (B)

Verified step by step guidance

1. First, recognize that the expression involves simplifying a square root of a negative number, which introduces imaginary numbers. Recall that the square root of -1 is represented by the imaginary unit 'i'.
2. Next, simplify the expression \( \sqrt{-75} \). This can be rewritten as \( \sqrt{75} \cdot \sqrt{-1} \), which becomes \( \sqrt{75} \cdot i \).
3. Factor 75 into its prime factors: \( 75 = 3 \times 5^2 \). This allows us to simplify \( \sqrt{75} \) to \( \sqrt{3} \times \sqrt{5^2} = \sqrt{3} \times 5 \).
4. Combine the simplified square root with the imaginary unit: \( 5 \sqrt{3} \cdot i \).
5. Thus, the simplified form of \( \sqrt{-75} \) is \( 5i \sqrt{3} \).