Step-by-step explanation:
a)
First, when simplifying radicals, we prime factorize the number within the radical. In this case, it's [tex]2^{5}[/tex]. For square roots, when we have 2 of the same prime number, it goes outside the radical. However, when it goes outside the radical, we only write that number once.
[tex]\sqrt{32} \\=\sqrt{2*2*2*2*2} \\=2*2\sqrt{2} \\=4\sqrt{2}[/tex]
b)
[tex]\sqrt{125x^{2}y^{7} } \\=\sqrt{5*5*5*x*x*y*y*y*y*y*y*y}\\=5*x*y*y*y\sqrt{5y}\\=5xy^{3}\sqrt{5y}[/tex]
c)
For this, we want the cubic root, so we need at least 3 of a number for it to go outside the radical.
[tex]\sqrt[3]{24x^{3}y^{8} } \\=\sqrt[3]{2*2*2*3*x*x*x*y*y*y*y*y*y*y*y} \\=2*x*y*y\sqrt[3]{3*y*y} \\=2xy^{2} \sqrt[3]{3y^{2} }[/tex]
