Question

Simplify. Assume all variables are positive.
((z^2/3)*(z^7/3))/(z^7/3)

Answer 1

To simplify the expression \(\frac{(z^{2/3})(z^{7/3})}{z^{7/3}}\), we can follow these steps:

1. First, simplify the numerator:
\[
z^{2/3} \cdot z^{7/3} = z^{(2/3 + 7/3)} = z^{9/3} = z^3
\]

2. Now we have:
\[
\frac{z^3}{z^{7/3}}
\]

3. Next, apply the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\):
\[
z^3 \cdot z^{-7/3} = z^{3 - 7/3}
\]

4. To subtract the exponents, convert 3 to a fraction with a common denominator:
\[
3 = \frac{9}{3}
\]
So,
\[
3 - \frac{7}{3} = \frac{9}{3} - \frac{7}{3} = \frac{2}{3}
\]

5. Therefore:
\[
z^{3 - 7/3} = z^{2/3}
\]

Thus, the simplified expression is:
\[
\boxed{z^{2/3}}
\]