Question

express the following fraction in simplest form using only positive elements

2(r^-2)^5/2r^-6

Answer 1

To simplify the expression \(\frac{2(r^{-2})^5}{2r^{-6}}\), we can follow these steps:

1. **Simplify the numerator**:
The numerator is \(2(r^{-2})^5\). We can simplify this as follows:
\[
(r^{-2})^5 = r^{-10}
\]
Therefore:
\[
2(r^{-2})^5 = 2r^{-10}
\]

2. **Substituting back into the fraction**:
The expression now looks like:
\[
\frac{2r^{-10}}{2r^{-6}}
\]

3. **Cancel out the common factor of 2**:
Since both the numerator and the denominator have a factor of 2, we can cancel these out:
\[
\frac{r^{-10}}{r^{-6}}
\]

4. **Simplifying the fraction**:
We can use the property \(\frac{a^m}{a^n} = a^{m-n}\):
\[
r^{-10 - (-6)} = r^{-10 + 6} = r^{-4}
\]

5. **Expressing in positive terms**:
To express \(r^{-4}\) in terms of positive exponents, we rewrite it as:
\[
r^{-4} = \frac{1}{r^4}
\]

Thus, the simplified form of the given expression is:
\[
\frac{1}{r^4}
\]