Question

Express the following fraction in simplest form, only using positive exponents.
start fraction, 2, w, to the power minus 1 , divided by, left parenthesis, minus, 3, w, to the power minus 3 , right parenthesis, to the power 4 , end fraction
(−3w
−3
)
4

2w
−1

​

Answer 1

To simplify the fraction \(\frac{2w^{-1}}{(-3w^{-3})^4}\), we will first simplify the denominator and then the entire fraction.

1. **Simplify the Denominator:**

The denominator is \((-3w^{-3})^4\). We can apply the exponent to both the constant and the variable:
\[
(-3)^4 \cdot (w^{-3})^4 = 81 \cdot w^{-12}
\]

Therefore, the denominator becomes:
\[
81w^{-12}
\]

2. **Rewrite the Fraction:**

Now we rewrite the original fraction with the simplified denominator:
\[
\frac{2w^{-1}}{81w^{-12}}
\]

3. **Dividing the Fractions:**

When dividing fractions, we can subtract the exponents of the same base:
\[
\frac{2}{81} \cdot w^{-1 - (-12)} = \frac{2}{81} \cdot w^{11}
\]

This gives us:
\[
\frac{2w^{11}}{81}
\]

4. **Result in Simplest Form:**

Since we want to express everything in positive exponents, we write the final answer as:
\[
\frac{2w^{11}}{81}
\]

Thus, the simplified form of the given fraction is:
\[
\frac{2w^{11}}{81}
\]