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. Recognize that the expression involves the square root of a negative number: \(\sqrt{-75}\). Since the square root of a negative number is not a real number, we use the imaginary unit \(i\), where \(i = \sqrt{-1}\).
2. Rewrite the square root of the negative number by separating the negative sign: \(\sqrt{-75} = \sqrt{-1 \times 75} = \sqrt{-1} \times \sqrt{75} = i \sqrt{75}\).
3. Simplify the square root of 75 by factoring it into perfect squares: \$75 = 25 \times 3\(, so \)\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3}$.
4. Calculate the square root of the perfect square: \(\sqrt{25} = 5\), so \(\sqrt{75} = 5 \sqrt{3}\).
5. Combine all parts to write the simplified form: \(\sqrt{-75} = i \times 5 \sqrt{3} = 5i \sqrt{3}\).