Question

Express the following fraction in simplest form, only using positive exponents.
start fraction, left parenthesis, 2, v, to the power 4 , right parenthesis, cubed, divided by, minus, 6, v, to the power minus 2 , end fraction
−6v
−2

(2v
4
)
3

​

Answer 1

To simplify the fraction \(\frac{(2v^4)^3}{-6v^{-2}}\), follow these steps:

1. **Simplify the numerator:**
\[
(2v^4)^3 = 2^3 (v^4)^3 = 8v^{12}
\]

2. **Write the expression now:**
\[
\frac{8v^{12}}{-6v^{-2}}
\]

3. **Combine the coefficients:**
\[
\frac{8}{-6} = -\frac{4}{3}
\]
So now we have:
\[
-\frac{4}{3} \cdot \frac{v^{12}}{v^{-2}}
\]

4. **Use the property of exponents to simplify \( \frac{v^{12}}{v^{-2}} \):**
\[
\frac{v^{12}}{v^{-2}} = v^{12 - (-2)} = v^{12 + 2} = v^{14}
\]

5. **Combine everything together:**
\[
-\frac{4}{3} v^{14}
\]

Thus, the expression in simplest form, with only positive exponents, is:
\[
-\frac{4}{3} v^{14}
\]