$a)$

$\sqrt{32}=\sqrt{16\cdot 2}=\sqrt{16}\cdot \sqrt{2}=4\sqrt{2}$

$\sqrt{75}=\sqrt{25\cdot 3}=\sqrt{25}\cdot \sqrt{3}=5\sqrt{3}$

$\sqrt{40}=\sqrt{4\cdot 10}=\sqrt{4}\cdot \sqrt{10}=2\sqrt{10}$

 

$b)$

$\sqrt{200}=\sqrt{100\cdot 2}=\sqrt{100}\cdot \sqrt{2}=10\sqrt{2}$ $\sqrt{200}=\sqrt{100\cdot 2}=\sqrt{100}\cdot \sqrt{2}=10\sqrt{2}$

$\sqrt{108}=\sqrt{36\cdot 3}=\sqrt{36}\cdot \sqrt{3}=6\sqrt{3}$

$\sqrt{810}=\sqrt{81\cdot 10}=\sqrt{81}\cdot \sqrt{10}=9\sqrt{10}$

 

$c)$

$\sqrt{\frac{3}{4}}=\sqrt{\frac{1}{4}\cdot 3}=\sqrt{\frac{1}{4}}\cdot \sqrt{3}=\frac{1}{2}\sqrt{3}$

$\sqrt{\frac{2}{25}}=\sqrt{\frac{1}{25}\cdot 2}=\sqrt{\frac{1}{25}}\cdot \sqrt{2}=\frac{1}{5}\sqrt{2}$

$\sqrt{\frac{5}{49}}=\sqrt{\frac{1}{49}\cdot 5}=\sqrt{\frac{1}{49}}\cdot \sqrt{5}=\frac{1}{7}\sqrt{5}$

 

$d)$

$\sqrt[3]{40}=\sqrt[3]{8\cdot 5}=\sqrt[3]{8}\cdot \sqrt[3]{5}=2\sqrt[3]{5}$

$\sqrt[3]{96}=\sqrt[3]{8\cdot 12}=\sqrt[3]{8}\cdot \sqrt[3]{12}=2\sqrt[3]{12}$

$\sqrt[3]{243}=\sqrt[3]{27\cdot 9}=\sqrt[3]{27}\cdot \sqrt[3]{9}=3\sqrt[3]{9}$  

 

$e)$

$\sqrt[3]{-24}=\sqrt[3]{{\left(-1\right)}^3\cdot 3\cdot 8}=-1\sqrt[3]{3}\cdot \sqrt[3]{8}=-2\sqrt[3]{3}$

$\sqrt[3]{-250}=\sqrt[3]{{\left(-1\right)}^3\cdot 125\cdot 2}=-1\sqrt[3]{2}\cdot \sqrt[3]{125}=-5\sqrt[3]{2}$

$\sqrt[3]{-128}=\sqrt[3]{{\left(-1\right)}^3\cdot 8\cdot 16}=-1\sqrt[3]{16}\cdot \sqrt[3]{8}=-2\sqrt[3]{16}$

 

$f)$

$\sqrt[3]{\frac{5}{27}}=\sqrt[3]{5\cdot \frac{1}{27}}=\sqrt[3]{5}\cdot \sqrt[3]{\frac{1}{27}}=\frac{1}{3}\sqrt[3]{5}$

$\sqrt[3]{\frac{7}{64}}=\sqrt[3]{7\cdot \frac{1}{64}}=\sqrt[3]{7}\cdot \sqrt[3]{\frac{1}{64}}=\frac{1}{4}\sqrt[3]{7}$

$\sqrt[3]{\frac{11}{125}}=\sqrt[3]{11\cdot \frac{1}{125}}=\sqrt[3]{11}\cdot \sqrt[3]{\frac{1}{125}}=\frac{1}{5}\sqrt[3]{11}$