Question

express the following fraction in simplest form using only positive elements

(4q^5)^3/4q^7

Answer 1

To simplify the expression \(\frac{(4q^5)^3}{4q^7}\), we start by expanding the numerator:

1. **Expand the numerator:**
\[
(4q^5)^3 = 4^3 \cdot (q^5)^3 = 64q^{15}
\]

2. **Rewrite the fraction:**
\[
\frac{(4q^5)^3}{4q^7} = \frac{64q^{15}}{4q^7}
\]

3. **Simplify the coefficient:**
\[
\frac{64}{4} = 16
\]

4. **Simplify the \(q\) terms using the property \(\frac{a^m}{a^n} = a^{m-n}\):**
\[
\frac{q^{15}}{q^{7}} = q^{15-7} = q^8
\]

5. **Combine the results:**
\[
\frac{(4q^5)^3}{4q^7} = 16q^8
\]

Thus, the fraction \(\frac{(4q^5)^3}{4q^7}\) in simplest form is:

\[
\boxed{16q^8}
\]