sqrt[3]((2.5)/(sqrt(5))) times sqrt(52 times 3)

To simplify the expression \(\sqrt{\left(\frac{8.5}{\sqrt{5}}\right) \times \sqrt{52 \times 3}}\), we can follow these steps:<br /><br />1. Simplify the expression inside the square root:<br /> \[<br /> \left(\frac{8.5}{\sqrt{5}}\right) \times \sqrt{52 \times 3}<br /> \]<br /><br />2. Simplify \(\frac{8.5}{\sqrt{5}}\):<br /> \[<br /> \frac{8.5}{\sqrt{5}} = \frac{8.5 \sqrt{5}}{5}<br /> \]<br /><br />3. Simplify \(\sqrt{52 \times 3}\):<br /> \[<br /> \sqrt{52 \times 3} = \sqrt{156}<br /> \]<br /><br />4. Combine the simplified parts:<br /> \[<br /> \left(\frac{8.5 \sqrt{5}}{5}\right) \times \sqrt{156}<br /> \]<br /><br />5. Multiply the terms:<br /> \[<br /> \frac{8.5 \sqrt{5} \times \sqrt{156}}{5} = \frac{8.5 \sqrt{780}}{5}<br /> \]<br /><br />6. Simplify \(\sqrt{780}\):<br /> \[<br /> \sqrt{780} = \sqrt{4 \times 195} = 2 \sqrt{195}<br /> \]<br /><br />7. Substitute back:<br /> \[<br /> \frac{8.5 \times 2 \sqrt{195}}{5} = \frac{17 \sqrt{195}}{5}<br /> \]<br /><br />8. Finally, take the square root of the entire expression:<br /> \[<br /> \sqrt{\frac{17 \sqrt{195}}{5}} = \frac{\sqrt{17 \times 195}}{5} = \frac{\sqrt{3315}}{5}<br /> \]<br /><br />So, the simplified form of the given expression is:<br />\[<br />\boxed{\frac{\sqrt{3315}}{5}}<br />\]