Sure, let's simplify the expression [tex]\(\sqrt{3 r^5 s^3} \cdot \sqrt{35}\)[/tex] step by step.
1. Combine the Radicals:
We start by combining the two square roots into a single square root. By the property of square roots, [tex]\( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \)[/tex], we have:
[tex]\[
\sqrt{3 r^5 s^3} \cdot \sqrt{35} = \sqrt{(3 r^5 s^3) \cdot 35}
\][/tex]
2. Multiply Inside the Radical:
Next, we multiply the terms inside the square root:
[tex]\[
(3 r^5 s^3) \cdot 35 = 3 \cdot 35 \cdot r^5 \cdot s^3
\][/tex]
3. Simplify the Product:
Simplify the constants (numbers) inside the radical:
[tex]\[
3 \cdot 35 = 105
\][/tex]
Therefore, the expression now becomes:
[tex]\[
\sqrt{105 r^5 s^3}
\][/tex]
4. Rewrite the Expression:
We can rewrite [tex]\(\sqrt{105 r^5 s^3}\)[/tex] as:
[tex]\[
\sqrt{105 r^5 s^3} = \sqrt{105} \cdot \sqrt{r^5 s^3}
\][/tex]
So, by simplifying the given expression, we get:
[tex]\[
\sqrt{105} \cdot \sqrt{r^5 s^3}
\][/tex]
Therefore, the simplified form of the expression [tex]\(\sqrt{3 r^5 s^3} \cdot \sqrt{35}\)[/tex] is:
[tex]\[
\sqrt{105} \cdot \sqrt{r^5 s^3}
\][/tex]
