Question

Express the following fraction in simplest form using only positive exponents.
start fraction, 4, z, to the power 4 , divided by, 3, left parenthesis, z, to the power 5 , right parenthesis, squared, end fraction
3(z
5
)
2

4z
4

​

Answer 1

To simplify the fraction \(\frac{4z^4}{3(z^5)^2}\), follow these steps:

1. **Square the expression in the denominator**:
\[
(z^5)^2 = z^{10}
\]
So, the denominator becomes \(3z^{10}\).

2. **Rewrite the fraction**:
\[
\frac{4z^4}{3z^{10}}
\]

3. **Separate the constants and the variables**:
\[
\frac{4}{3} \cdot \frac{z^4}{z^{10}}
\]

4. **Apply the exponent rule** (subtract the exponents for the same base):
\[
\frac{z^4}{z^{10}} = z^{4-10} = z^{-6}
\]

5. **Combine the results**:
\[
\frac{4}{3} z^{-6}
\]

6. **Convert to positive exponents**:
\[
\frac{4}{3} z^{-6} = \frac{4}{3} \cdot \frac{1}{z^6} = \frac{4}{3z^6}
\]

Thus, the simplified form of the fraction is:
\[
\frac{4}{3z^6}
\]