<p>If <em>a</em> < 0, <em>b</em> > 0, then \(\sqrt{a} \cdot \sqrt{b}\) is equal to</p>
Step-by-Step Solution
Key Concept: When dealing with square roots of negative numbers, we must use the imaginary unit i = √(-1). For a < 0, √a = i√|a|, so √a · √b = i√|a| · √b = i√(|a|b), NOT √(ab).
<p><strong>Step 1:</strong> Since a < 0, write a = -|a|, so √a = √(-|a|) = √(-1) · √|a| = i√|a|</p><p><strong>Step 2:</strong> Since b > 0, we have √b = √b (standard real square root)</p><p><strong>Step 3:</strong> Multiply them: √a · √b = i√|a| · √b = i√(|a|b)</p><p><strong>Step 4:</strong> Since a < 0 and b > 0, we have |a|b = -ab (because |a| = -a), so: √a · √b = i√(-ab) or equivalently √(ab) is undefined in reals, but the principal value is i√(-ab)</p><p>∴ Answer: B (typically written as i√(-ab) or √(-ab) depending on options)</p>
Correct Answer: B