Question

Express the following fraction in simplest form using only positive exponents.
start fraction, 3, left parenthesis, s, cubed, right parenthesis, to the power 4 , divided by, 12, s, to the power 8 , end fraction
12s
8

3(s
3
)
4

​

Answer 1

To simplify the fraction \(\frac{3(s^3)^4}{12s^8}\), we can follow these steps:

1. **Simplify the numerator**:
\[
(s^3)^4 = s^{3 \times 4} = s^{12}
\]
So the numerator becomes:
\[
3s^{12}
\]

2. **Rewrite the fraction**:
\[
\frac{3s^{12}}{12s^8}
\]

3. **Divide the coefficients**:
\[
\frac{3}{12} = \frac{1}{4}
\]

4. **Simplify the terms with the same base** (using the rule \(\frac{a^m}{a^n} = a^{m-n}\)):
\[
\frac{s^{12}}{s^8} = s^{12-8} = s^4
\]

5. **Combine the results**:
\[
\frac{3s^{12}}{12s^8} = \frac{1}{4} s^4
\]

Thus, the simplified form of the fraction \(\frac{3(s^3)^4}{12s^8}\) is:
\[
\frac{1}{4}s^4
\]
or simply:
\[
\frac{s^4}{4}
\]

This represents the expression in simplest form using only positive exponents.