https://file.aiquickdraw.com/mathbot/file/1726886879962o66ondbf.jpg

The given expression is:

\[
\sqrt{14r^2s} \cdot \sqrt{35}
\]

### Step 1: Combine the square roots.
We can multiply the contents of the square roots together under one square root:

\[
\sqrt{14r^2s \cdot 35} = \sqrt{14 \cdot 35 \cdot r^2 \cdot s}
\]

### Step 2: Simplify the multiplication.
Multiply 14 and 35:

\[
14 \cdot 35 = 490
\]

So the expression becomes:

\[
\sqrt{490r^2s}
\]

### Step 3: Factor 490 to simplify the square root.
Factor 490 as:

\[
490 = 49 \cdot 10
\]

Thus:

\[
\sqrt{490r^2s} = \sqrt{49 \cdot 10 \cdot r^2 \cdot s}
\]

### Step 4: Simplify the square root.
We can take the square root of the perfect squares \(49\) and \(r^2\):

\[
\sqrt{49} = 7 \quad \text{and} \quad \sqrt{r^2} = r
\]

This gives:

\[
7r \cdot \sqrt{10s}
\]

### Final Answer:
The simplified expression is:

\[
7r\sqrt{10s}
\]

Would you like further details or have any questions?

Here are 5 related questions for further practice:
1. How would the expression change if there was a different power on \( r \)?
2. Can you multiply two different square roots with different exponents?
3. How do you simplify square roots with variables raised to powers higher than 2?
4. What would the result be if \( r \) and \( s \) were not positive or zero?
5. How do you handle cube roots or higher-order roots in such expressions?

**Tip:** When simplifying square roots, always check for any perfect square factors, as they can be simplified easily.

Multiply. Assume r and s are greater than or equal to zero, and write your answer in simplest form: sqrt(14r^2s) * sqrt(35).

Learn how to multiply and simplify the square root expression sqrt(14r^2s) * sqrt(35) in this detailed step-by-step guide. Understand key concepts like radical multiplication and simplification. Suitable for students in Grades 9-11.

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