Let's break down and solve the given expression step by step:
Given expression:
[tex]\[ \sqrt{3 \times 5^{-3}} \div \sqrt[3]{3^{-1}} \cdot \sqrt{5} \times \sqrt[6]{3 \times 5^4} \][/tex]
1. Calculate [tex]\(\sqrt{3 \times 5^{-3}}\)[/tex]:
[tex]\[ 3 \times 5^{-3} = 3 \times \frac{1}{5^3} = 3 \times \frac{1}{125} = \frac{3}{125} \][/tex]
[tex]\[ \sqrt{\frac{3}{125}} \approx 0.15491933384829668 \][/tex]
2. Calculate [tex]\(\sqrt[3]{3^{-1}}\)[/tex]:
[tex]\[ 3^{-1} = \frac{1}{3} \][/tex]
[tex]\[ \sqrt[3]{\frac{1}{3}} \approx 0.6933612743506347 \][/tex]
3. Calculate [tex]\(\sqrt{5}\)[/tex]:
[tex]\[ \sqrt{5} \approx 2.23606797749979 \][/tex]
4. Calculate [tex]\(\sqrt[6]{3 \times 5^4}\)[/tex]:
[tex]\[ 3 \times 5^4 = 3 \times 625 = 1875 \][/tex]
[tex]\[ \sqrt[6]{1875} \approx 3.5115609594099815 \][/tex]
Let's combine these intermediate results into the given expression:
[tex]\[ \frac{\sqrt{3 \times 5^{-3}}}{\sqrt[3]{3^{-1}}} \cdot \sqrt{5} \times \sqrt[6]{3 \times 5^4} \approx \frac{0.15491933384829668}{0.6933612743506347} \cdot 2.23606797749979 \cdot 3.5115609594099815 \][/tex]
5. Perform the division:
[tex]\[ \frac{0.15491933384829668}{0.6933612743506347} \approx 0.223453664983188 \][/tex]
6. Perform the remaining multiplications:
[tex]\[ 0.223453664983188 \cdot 2.23606797749979 \approx 0.49981765609582 \][/tex]
[tex]\[ 0.49981765609582 \cdot 3.5115609594099815 \approx 1.75441064292772 \][/tex]
Hence, the final result is approximately:
[tex]\[ 1.75441064292772 \][/tex]
So, the step-by-step solution to the given expression is:
[tex]\[ \sqrt{3 \times 5^{-3}} \div \sqrt[3]{3^{-1}} \cdot \sqrt{5} \times \sqrt[6]{3 \times 5^4} \approx 1.75441064292772 \][/tex]
