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 square root of a negative number can be expressed using the imaginary unit \(i\), where \(i = \sqrt{-1}\). So rewrite \(\sqrt{-75}\) as \(\sqrt{-1 \times 75} = \sqrt{-1} \times \sqrt{75} = i \sqrt{75}\).
2. Next, simplify the square root of 75 by factoring it into its prime factors or perfect squares. Since \$75 = 25 \times 3\(, rewrite \)\sqrt{75}\( as \)\sqrt{25 \times 3}$.
3. Use the property of square roots that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to separate the factors: \(\sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3}\).
4. Calculate the square root of the perfect square \$25\(, which is \)5\(, so \)\sqrt{25} = 5\(. Now the expression becomes \)i \times 5 \times \sqrt{3}$.
5. Finally, write the simplified form as \$5i \sqrt{3}\(, which is the simplified form of \)\sqrt{-75}$.