Let's break down the expression step by step and simplify it.
Given expression:
[tex]\[
\left(\frac{3}{4} \sqrt{15}\right)\left(16 \sqrt{5}\right)
\][/tex]
1. Separate the constants and radicals:
[tex]\[
\left(\frac{3}{4}\right) \cdot 16 \quad \text{and} \quad \sqrt{15} \cdot \sqrt{5}
\][/tex]
2. Simplify the constants:
[tex]\[
\left(\frac{3}{4}\right) \cdot 16 = \frac{3 \cdot 16}{4} = \frac{48}{4} = 12
\][/tex]
3. Combine the radicals:
[tex]\[
\sqrt{15} \cdot \sqrt{5} = \sqrt{15 \times 5} = \sqrt{75}
\][/tex]
4. Simplify the radical [tex]\(\sqrt{75}\)[/tex]:
[tex]\[
\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \cdot \sqrt{3} = 5 \sqrt{3}
\][/tex]
5. Combine the simplified constants and radicals:
[tex]\[
12 \cdot 5 \sqrt{3} = 60 \sqrt{3}
\][/tex]
Thus, the simplified expression is:
[tex]\[
60 \sqrt{3}
\][/tex]
To summarize:
- Combine and simplify the constants,
- Combine the radicals and simplify,
- Finally, multiply the simplified constants and radicals to get the final result.
Thus, the answer is [tex]\( 60 \sqrt{3} \)[/tex].
