Option C:
[tex]$\frac{\sqrt[3]{100 x}}{5}=\sqrt[3]{\frac{4 x}{5}}[/tex]
Solution:
Given expression is
[tex]$\sqrt[3]{\frac{4 x}{5}}[/tex]
Note: [tex]\sqrt[3]{125}=\sqrt[3]{{5^3}} = 5[/tex]
To find the correct expression for the above simplified expression.
Option A: [tex]\frac{\sqrt[3]{4 x}}{5}[/tex]
5 can be written as [tex]\sqrt[3]{125}[/tex].
[tex]$\frac{\sqrt[3]{4 x}}{5}=\frac{\sqrt[3]{4 x}}{\sqrt[3]{125} }[/tex]
[tex]$=\sqrt[3]{\frac{4x}{125} }[/tex]
It is not the given simplified expression.
Option B: [tex]\frac{\sqrt[3]{20 x}}{5}[/tex]
[tex]$\frac{\sqrt[3]{20 x}}{5}=\frac{\sqrt[3]{20 x}}{\sqrt[3]{125} }[/tex]
[tex]$=\sqrt[3]{\frac{20x}{125} }[/tex]
Cancel the common factor in both numerator and denominator.
[tex]$=\sqrt[3]{\frac{4x}{25} }[/tex]
It is not the given simplified expression.
Option C: [tex]\frac{\sqrt[3]{100 x}}{5}[/tex]
[tex]$\frac{\sqrt[3]{100 x}}{5}=\frac{\sqrt[3]{100 x}}{\sqrt[3]{125} }[/tex]
[tex]$=\sqrt[3]{\frac{100x}{125} }[/tex]
Cancel the common factor in both numerator and denominator.
[tex]$=\sqrt[3]{\frac{4 x}{5}}[/tex]
It is the given simplified expression.
Option D: [tex]\frac{\sqrt[3]{100 x}}{125}[/tex]
[tex]$\frac{\sqrt[3]{100 x}}{125}=\frac{\sqrt[3]{100 x}}{5^3}[/tex]
It is not the given simplified expression.
Hence Option C is the correct answer.
[tex]$\frac{\sqrt[3]{100 x}}{5}=\sqrt[3]{\frac{4 x}{5}}[/tex]
