Answer :
Answer:
We conclude that:
[tex]\left(\:\:\frac{x^4}{\frac{3x^2}{3}}\right)^{\frac{1}{3}}\:=\:x^{\frac{2}{3}}[/tex]
Hence, option A i.e. [tex]x^{\frac{2}{3}}[/tex] is true.
Step-by-step explanation:
Given the expression
[tex]\left(\:\:\frac{x^4}{\frac{3x^2}{3}}\right)^{\frac{1}{3}}[/tex]
Apply exponent rule: [tex]\left(\frac{a}{b}\right)^c=\frac{a^c}{b^c}[/tex]
[tex]\left(\frac{x^4}{\frac{3x^2}{3}}\right)^{\frac{1}{3}}=\frac{\left(x^4\right)^{\frac{1}{3}}}{\left(\frac{3x^2}{3}\right)^{\frac{1}{3}}}[/tex]
[tex]=\frac{x^{\frac{4}{3}}}{\left(\frac{3x^2}{3}\right)^{\frac{1}{3}}}[/tex] ∵ [tex]\:\:\:\left(x^4\right)^{\frac{1}{3}}=x^{\frac{4}{3}}[/tex]
[tex]=\frac{x^{\frac{4}{3}}}{\frac{\left(3x^2\right)^{\frac{1}{3}}}{3^{\frac{1}{3}}}}[/tex]
[tex]=\frac{x^{\frac{4}{3}}}{x^{\frac{2}{3}}}[/tex]
Apply exponent rule: [tex]\frac{x^a}{x^b}=x^{a-b}[/tex]
[tex]=x^{\frac{4}{3}-\frac{2}{3}}[/tex]
[tex]=x^{\frac{2}{3}}\\[/tex]
Therefore, we conclude that:
[tex]\left(\:\:\frac{x^4}{\frac{3x^2}{3}}\right)^{\frac{1}{3}}\:=\:x^{\frac{2}{3}}[/tex]
Hence, option A i.e. [tex]x^{\frac{2}{3}}[/tex] is true.