Question

Express the following fraction in simplest form, only using positive exponents.
start fraction, 10, j, to the power minus 3 , divided by, minus, 2, left parenthesis, j, to the power minus 3 , right parenthesis, cubed, end fraction
−2(j
−3
)
3

10j
−3

​

Answer 1

To simplify the expression \(\frac{10 j^{-3}}{-2 (j^{-3})^3}\), we will follow these steps:

1. **Simplify the denominator**:
\[
-2 (j^{-3})^3 = -2 j^{-9}
\]
This is because when you raise a power to another power, you multiply the exponents: \((-3) \cdot 3 = -9\).

2. **Rewrite the fraction**:
So, we rewrite the fraction:
\[
\frac{10 j^{-3}}{-2 j^{-9}}
\]

3. **Divide the numerical coefficients**:
The numerical part is:
\[
\frac{10}{-2} = -5
\]

4. **Divide the variable parts** (using the property \(\frac{a^m}{a^n} = a^{m-n}\)):
For the variable part, we have:
\[
\frac{j^{-3}}{j^{-9}} = j^{-3 - (-9)} = j^{-3 + 9} = j^{6}
\]

5. **Combine the results**:
Therefore, the expression simplifies to:
\[
-5 j^{6}
\]

6. **Express in positive exponents**:
Since the original expression does not have any negative base after simplification, we keep it as is.

So, the final result in simplest form using only positive exponents is:
\[
-5 j^{6}
\]