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

Question

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:

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