125 = 5³, and the given expression reduces to
[tex]\left(\sqrt{125}\right)^{\frac13}\cdot\left(\sqrt[3]{125}\right)^{\frac12}=\left(125^{\frac12}\right)^{\frac13}\cdot\left(125^{\frac13}\right)^{\frac12}=125^{\frac16}\cdot125^{\frac16}=125^{\frac13}[/tex]
so it has a value of 5.
Now:
• the first choice is equivalent, since 625 = 25², so
[tex]\dfrac{\sqrt[3]{625}}{5^{\frac13}}=\dfrac{25^{\frac23}}{5^{\frac13}}=\dfrac{5^{\frac43}}{5^{\frac13}}=5[/tex]
• the second choice is not equivalent, since
[tex]5 \cdot 25^{\frac13} = 5 \cdot 5^{\frac23} = 5^{\frac43}[/tex]
• the third choice is also equivalent, since
[tex]\dfrac{\left(\sqrt[4]{5}\right)^7}{125^{\frac14}}=\dfrac{5^{\frac74}}{5^{\frac34}}=5[/tex]
• the fourth choice is also equivalent, since
[tex]\dfrac{5^{\frac12}\cdot25^{\frac13}}{\sqrt[6]{5}}=\dfrac{5^{\frac12}\cdot5^{\frac23}}{5^{\frac16}}=\dfrac{5^{\frac76}}{5^{\frac16}}=5[/tex]
