Simplify t he ex pres sion. sqrt [3](375u^7v^8)= square

To simplify the expression \(\sqrt[3]{375u^{7}v^{8}}\), we need to break down each component inside the cube root and then apply the cube root to each part separately.<br /><br />First, let's factorize 375:<br />\[ 375 = 3 \times 5^3 \]<br /><br />Next, consider the variables:<br />\[ u^7 \]<br />\[ v^8 \]<br /><br />Now, we apply the cube root to each part:<br />\[ \sqrt[3]{375u^7v^8} = \sqrt[3]{3 \times 5^3 \times u^7 \times v^8} \]<br /><br />We can separate the cube root into individual parts:<br />\[ \sqrt[3]{3} \times \sqrt[3]{5^3} \times \sqrt[3]{u^7} \times \sqrt[3]{v^8} \]<br /><br />Evaluate each cube root:<br />\[ \sqrt[3]{3} \text{ remains as } \sqrt[3]{3} \]<br />\[ \sqrt[3]{5^3} = 5 \]<br />\[ \sqrt[3]{u^7} = u^{7/3} \]<br />\[ \sqrt[3]{v^8} = v^{8/3} \]<br /><br />Putting it all together:<br />\[ \sqrt[3]{375u^7v^8} = \sqrt[3]{3} \times 5 \times u^{7/3} \times v^{8/3} \]<br /><br />So, the simplified expression is:<br />\[ 5 \sqrt[3]{3} u^{7/3} v^{8/3} \]