Sure, let's break down the expression step-by-step using the appropriate algebraic rules for simplifying square roots and fractions.
We need to simplify the expression:
[tex]\[
\frac{\sqrt{15 x^6}}{\sqrt{5 x^5}} \cdot \sqrt{3 x^3}
\][/tex]
### Step 1: Simplify the Square Roots
1. Simplify [tex]\(\sqrt{15 x^6}\)[/tex]:
[tex]\[
\sqrt{15x^6} = \sqrt{15} \cdot \sqrt{x^6}
\][/tex]
Since [tex]\(\sqrt{x^6} = x^3\)[/tex], we get:
[tex]\[
\sqrt{15 x^6} = \sqrt{15} \cdot x^3
\][/tex]
2. Simplify [tex]\(\sqrt{5 x^5}\)[/tex]:
[tex]\[
\sqrt{5 x^5} = \sqrt{5} \cdot \sqrt{x^5}
\][/tex]
Since [tex]\(\sqrt{x^5} = x^{5/2}\)[/tex], we get:
[tex]\[
\sqrt{5 x^5} = \sqrt{5} \cdot x^{5/2}
\][/tex]
### Step 2: Divide the Numerator by the Denominator
Now, we divide [tex]\(\sqrt{15 x^6}\)[/tex] by [tex]\(\sqrt{5 x^5}\)[/tex]:
[tex]\[
\frac{\sqrt{15 x^6}}{\sqrt{5 x^5}} = \frac{\sqrt{15} \cdot x^3}{\sqrt{5} \cdot x^{5/2}}
\][/tex]
We can simplify this further by separating the constants and the variables:
[tex]\[
\frac{\sqrt{15}}{\sqrt{5}} \cdot \frac{x^3}{x^{5/2}}
\][/tex]
Simplify [tex]\(\frac{\sqrt{15}}{\sqrt{5}}\)[/tex]:
[tex]\[
\frac{\sqrt{15}}{\sqrt{5}} = \sqrt{\frac{15}{5}} = \sqrt{3}
\][/tex]
Simplify [tex]\(\frac{x^3}{x^{5/2}}\)[/tex]:
[tex]\[
\frac{x^3}{x^{5/2}} = x^{3 - 5/2} = x^{6/2 - 5/2} = x^{1/2}
\][/tex]
Putting it together, we have:
[tex]\[
\frac{\sqrt{15 x^6}}{\sqrt{5 x^5}} = \sqrt{3} \cdot x^{1/2} = \sqrt{3x}
\][/tex]
### Step 3: Multiply by [tex]\(\sqrt{3 x^3}\)[/tex]
Now, multiply the result by [tex]\(\sqrt{3 x^3}\)[/tex]:
[tex]\[
\sqrt{3x} \cdot \sqrt{3 x^3}
\][/tex]
Combine the square roots:
[tex]\[
\sqrt{3x \cdot 3x^3} = \sqrt{9x^4}
\][/tex]
Simplify the square root:
[tex]\[
\sqrt{9 x^4} = 3 x^2
\][/tex]
### Final Answer
The simplified form of the expression [tex]\(\frac{\sqrt{15 x^6}}{\sqrt{5 x^5}} \cdot \sqrt{3 x^3}\)[/tex] is:
[tex]\[
3 x^2
\][/tex]
