Question

Which option lists am expression that is not equivalent to 4^2/3

Answer 1

remember that

$x^{\frac{a}{b}}=\sqrt[b]{x^a}$

also $x^{-a}=\frac{1}{x^a}$

and $(a^b)^c=a^{bc}$

so

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

first one, that one is clearly not equal since 0.25≠4

2nd one, 0.25=1/4, so $0.25^{\frac{-2}{3}}=(\frac{1}{4})^{\frac{-2}{3}=$  $\frac{1}{(\frac{1}{4})^{\frac{2}{3}}}=4^{\frac{2}{3}}$, which matches

3rd one, $\sqrt[3]{16}=\sqrt[3]{4^2}=4^{\frac{3}{2}}$, which matches

4th one $(\sqrt[3]{4})^2=(4^{\frac{1}{3}})^2=4^{\frac{2}{3}}$ which matches

answer is $0.25^{\frac{2}{3}}$