Question

Express the following fraction in simplest form using only positive exponents.
start fraction, left parenthesis, 3, r, squared, right parenthesis, cubed, divided by, 6, r, to the power 4 , end fraction
6r
4

(3r
2
)
3

​

Answer 1

To simplify the fraction \(\frac{(3r^2)^3}{6r^4}\), follow these steps:

1. **Simplify the numerator**:

\[
(3r^2)^3 = 3^3 \cdot (r^2)^3 = 27r^6
\]

2. **Write the fraction**:

\[
\frac{27r^6}{6r^4}
\]

3. **Simplify the coefficients**:

Divide 27 by 6:

\[
\frac{27}{6} = \frac{9}{2}
\]

(since \(27 = 9 \cdot 3\) and \(6 = 3 \cdot 2\), they share a common factor of 3).

4. **Simplify the variable part**:

Use the property of exponents for \(r\):

\[
\frac{r^6}{r^4} = r^{6-4} = r^2
\]

5. **Combine the results**:

Putting it all together, we have:

\[
\frac{27r^6}{6r^4} = \frac{9}{2} r^2
\]

Therefore, the fraction \(\frac{(3r^2)^3}{6r^4}\) in simplest form with only positive exponents is:

\[
\frac{9r^2}{2}
\]