Simplify.

To simplify the expression \(3\sqrt{5a}\cdot 8\sqrt{35a^2}\), we will first combine the coefficients and the square roots separately.

1. **Multiply the coefficients**:
\[
3 \cdot 8 = 24
\]

2. **Combine the square roots**:
\[
\sqrt{5a} \cdot \sqrt{35a^2} = \sqrt{(5a)(35a^2)} = \sqrt{175a^3}
\]

Now we compute \(175a^3\) to simplify the square root:
- **Factor \(175\)**:
\[
175 = 25 \cdot 7 = 5^2 \cdot 7
\]
Therefore,
\[
175a^3 = 25 \cdot 7 \cdot a^3 = 5^2 \cdot 7 \cdot a^2 \cdot a = (5^2)(7)(a^2)(a)
\]

3. **Simplify the square root**:
\[
\sqrt{175a^3} = \sqrt{5^2 \cdot 7 \cdot a^2 \cdot a} = \sqrt{5^2} \cdot \sqrt{7} \cdot \sqrt{a^2} \cdot \sqrt{a}
\]
This simplifies to:
\[
5 \cdot \sqrt{7} \cdot a \cdot \sqrt{a} = 5a\sqrt{7a}
\]

4. **Combine the results**:
Now we combine the coefficient \(24\) with the simplified square root:
\[
24 \sqrt{175a^3} = 24 \cdot 5a \sqrt{7a} = 120a \sqrt{7a}
\]

Thus, the final simplified expression is:
\[
\boxed{120a\sqrt{7a}}
\]