Express the following fraction in simplest form using only positive exponents.

Question

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: \[ \frac{4}{3z^6} \]