If `a,b in R` such that `ab gt 0`, then `sqrt(a)sqrt(b)` is equal to

Options

A. `sqrt(|a||b|)`
B. `-sqrt(|a||b|)`
C. `sqrt(ab)`
D. none of these

Explanation

To solve the problem, we need to analyze the expression \(\sqrt{a} \cdot \sqrt{b}\) under the condition that \(ab > 0\). This means that either both \(a\) and \(b\) are positive, or both \(a\) and \(b\) are negative.

### Step-by-Step Solution:

1. **Understanding the Condition**:
Since \(ab > 0\), we have two cases:
- Case 1: \(a > 0\) and \(b > 0\)
- Case 2: \(a < 0\) and \(b < 0\)

2. **Case 1: Both \(a\) and \(b\) are positive**:
- If \(a > 0\) and \(b > 0\), we can directly use the property of square roots:
\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \]
- Since both \(a\) and \(b\) are positive, \(ab\) is also positive, and thus \(\sqrt{ab}\) is defined.

3. **Case 2: Both \(a\) and \(b\) are negative**:
- If \(a < 0\) and \(b < 0\), we can express \(a\) and \(b\) in terms of their absolute values:
\[ a = -|a| \quad \text{and} \quad b = -|b| \]
- Therefore, we have:
\[ \sqrt{a} = \sqrt{-|a|} \quad \text{and} \quad \sqrt{b} = \sqrt{-|b|} \]
- This leads to:
\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{-|a|} \cdot \sqrt{-|b|} = \sqrt{(-1)(-|a|)(-|b|)} = \sqrt{(-1)^2 \cdot |a| \cdot |b|} = \sqrt{|a| \cdot |b|} \]
- Since both \(a\) and \(b\) are negative, \(|a|\) and \(|b|\) are positive, and thus \(\sqrt{|a| \cdot |b|}\) is also defined.

4. **Conclusion**:
In both cases, we find that:
\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \]
Therefore, the final answer is:
\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \]