Question

express the following fraction in simplest form using only positive elements

20x^-3c/(4x^4)^-5

Answer 1

To simplify the expression \( \frac{20x^{-3}c}{(4x^4)^{-5}} \), we can follow these steps:

1. **Simplify the denominator**: We know that \( (a^m)^{-n} = a^{-mn} \).
\[
(4x^4)^{-5} = 4^{-5} \cdot (x^4)^{-5} = \frac{1}{4^5} \cdot \frac{1}{x^{20}} = \frac{1}{1024} \cdot \frac{1}{x^{20}} = \frac{1}{1024x^{20}}
\]

2. **Replace the denominator in the expression**:
\[
\frac{20x^{-3}c}{(4x^4)^{-5}} = 20x^{-3}c \cdot 1024x^{20}
\]

3. **Multiply the numerators**:
\[
= 20 \cdot 1024 \cdot c \cdot x^{-3} \cdot x^{20}
\]
When multiplying powers of \( x \), we add the exponents:
\[
x^{-3} \cdot x^{20} = x^{-3+20} = x^{17}
\]

4. **Combine the constants**:
\[
20 \cdot 1024 = 20480
\]

5. **Putting it all together**:
\[
= 20480c \cdot x^{17}
\]

6. **Convert the expression back into a fraction**:
\[
= \frac{20480c x^{17}}{1}
\]

Putting it all together, the simplest form of the given expression is:

\[
\boxed{20480c x^{17}}
\]