Question

Which expression is equivalent to (x^4/3 x^2/3)^1/3

Answer 1

Answer:

We conclude that:

$\left(\:\:\frac{x^4}{\frac{3x^2}{3}}\right)^{\frac{1}{3}}\:=\:x^{\frac{2}{3}}$        

Hence, option A i.e. $x^{\frac{2}{3}}$  is true.    

Step-by-step explanation:

Given the expression

$\left(\:\:\frac{x^4}{\frac{3x^2}{3}}\right)^{\frac{1}{3}}$

Apply exponent rule:   $\left(\frac{a}{b}\right)^c=\frac{a^c}{b^c}$

$\left(\frac{x^4}{\frac{3x^2}{3}}\right)^{\frac{1}{3}}=\frac{\left(x^4\right)^{\frac{1}{3}}}{\left(\frac{3x^2}{3}\right)^{\frac{1}{3}}}$

             $=\frac{x^{\frac{4}{3}}}{\left(\frac{3x^2}{3}\right)^{\frac{1}{3}}}$        ∵ $\:\:\:\left(x^4\right)^{\frac{1}{3}}=x^{\frac{4}{3}}$

             $=\frac{x^{\frac{4}{3}}}{\frac{\left(3x^2\right)^{\frac{1}{3}}}{3^{\frac{1}{3}}}}$

             $=\frac{x^{\frac{4}{3}}}{x^{\frac{2}{3}}}$

Apply exponent rule:    $\frac{x^a}{x^b}=x^{a-b}$

             $=x^{\frac{4}{3}-\frac{2}{3}}$          

             $=x^{\frac{2}{3}}$

Therefore, we conclude that:

$\left(\:\:\frac{x^4}{\frac{3x^2}{3}}\right)^{\frac{1}{3}}\:=\:x^{\frac{2}{3}}$        

Hence, option A i.e. $x^{\frac{2}{3}}$  is true.