Question

simplify. assume all variables are positive.
\\(\frac{z^{\frac{7}{6}}}{z^{\frac{11}{6}}}\\)
write your answer in the form a or \\(\frac{a}{b}\\), where a and b are constants or variable expressions that have no variables in common. all exponents in your answer should be positive.

Explanation:

Step1: Use exponent rule for division

When dividing two powers with the same base $a^m\div a^n=a^{m - n}$. Here the base is $z$, $m=\frac{7}{6}$ and $n = \frac{11}{6}$. So $\frac{z^{\frac{7}{6}}}{z^{\frac{11}{6}}}=z^{\frac{7}{6}-\frac{11}{6}}$.

Step2: Calculate the exponent

$\frac{7}{6}-\frac{11}{6}=\frac{7 - 11}{6}=\frac{- 4}{6}=-\frac{2}{3}$. So we have $z^{-\frac{2}{3}}$.

Step3: Make the exponent positive

Using the rule $a^{-n}=\frac{1}{a^{n}}$, we get $z^{-\frac{2}{3}}=\frac{1}{z^{\frac{2}{3}}}$.

Answer:

$\frac{1}{z^{\frac{2}{3}}}$