Question

Express the following fraction in simplest form, only using positive exponents.
start fraction, 2, left parenthesis, a, to the power minus 1 , right parenthesis, cubed, divided by, minus, 10, a, to the power 5 , end fraction
−10a
5

2(a
−1
)
3

​

Answer 1

To simplify the given fraction

\[
\frac{2(a^{-1})^3}{-10a^5},
\]

we start by simplifying the numerator.

1. Let's simplify the numerator, \(2(a^{-1})^3\):

\[
(a^{-1})^3 = a^{-3},
\]

so

\[
2(a^{-1})^3 = 2a^{-3}.
\]

The fraction now looks like this:

\[
\frac{2a^{-3}}{-10a^5}.
\]

2. Next, we can simplify this fraction by dividing both the numerator and the denominator by 2:

\[
\frac{2a^{-3}}{-10a^5} = \frac{a^{-3}}{-5a^5}.
\]

3. Now the fraction is:

\[
\frac{a^{-3}}{-5a^5}.
\]

4. We can combine the exponents in the denominator:

\[
-5a^5 = -5 \cdot a^{5}.
\]

5. Thus the expression becomes:

\[
\frac{a^{-3}}{-5 \cdot a^{5}} = \frac{1}{-5} \cdot \frac{a^{-3}}{a^{5}}.
\]

6. Using the property of exponents, \(a^{-3} / a^5 = a^{-3-5} = a^{-8}\):

\[
\frac{1}{-5} \cdot a^{-8} = \frac{a^{-8}}{-5}.
\]

7. Since we need positive exponents, we can rewrite \(a^{-8}\) as \(\frac{1}{a^8}\):

\[
\frac{a^{-8}}{-5} = \frac{1}{-5a^8}.
\]

Thus, the final result in simplest form with positive exponents is:

\[
\frac{-1}{5a^8}.
\]