Question

fill in the blank so that the resulting statement is true.

the product rule for square roots states that if a and b are nonnegative, then \\(\sqrt{ab} = \underline{\quad\quad}\\).

options:
\\(\sqrt{a} \cdot \sqrt{b}\\)
\\(\sqrt{a} - \sqrt{b}\\)
\\(\sqrt{a} + \sqrt{b}\\)
\\(\frac{\sqrt{a}}{\sqrt{b}}\\)

Explanation:

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<reasoning>

Identify the given mathematical statement

The problem asks us to complete the statement of the product rule for square roots. We are given that \(a\) and \(b\) are nonnegative real numbers.

Recall the product rule for square roots

The product rule states that the square root of a product is equal to the product of the square roots of the individual factors. Mathematically, for any nonnegative real numbers \(a\) and \(b\):
\[
\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}
\]

Match with the provided options

Looking at the dropdown options in the image:

• \(\sqrt{a} \cdot \sqrt{b}\)
• \(\sqrt{a} - \sqrt{b}\)
• \(\sqrt{a} + \sqrt{b}\)
• \(\frac{\sqrt{a}}{\sqrt{b}}\)

The correct option that completes the statement is \(\sqrt{a} \cdot \sqrt{b}\).
</reasoning>

<answer>
Fill in the blank so that the resulting statement is true.

The product rule for square roots states that if a and b are nonnegative, then \(\sqrt{ab} =\) <blank>\(\sqrt{a} \cdot \sqrt{b}\)</blank>.
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Product Rule for Radicals"
]
}
</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Identify the given mathematical statement

The problem asks us to complete the statement of the product rule for square roots. We are given that \(a\) and \(b\) are nonnegative real numbers.

Recall the product rule for square roots

The product rule states that the square root of a product is equal to the product of the square roots of the individual factors. Mathematically, for any nonnegative real numbers \(a\) and \(b\):
\[
\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}
\]

Match with the provided options

Looking at the dropdown options in the image:

• \(\sqrt{a} \cdot \sqrt{b}\)
• \(\sqrt{a} - \sqrt{b}\)
• \(\sqrt{a} + \sqrt{b}\)
• \(\frac{\sqrt{a}}{\sqrt{b}}\)

The correct option that completes the statement is \(\sqrt{a} \cdot \sqrt{b}\).
</reasoning>

<answer>
Fill in the blank so that the resulting statement is true.

The product rule for square roots states that if a and b are nonnegative, then \(\sqrt{ab} =\) <blank>\(\sqrt{a} \cdot \sqrt{b}\)</blank>.
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Product Rule for Radicals"
]
}
</post_analysis>