To find the solution to [tex]\(\left(\frac{-4}{3}\right)^7 \times\left(\frac{-4}{3}\right)^8 \div\left(\frac{-3}{4}\right)^5\)[/tex], follow these steps:
1. Combine like bases using exponent rules:
[tex]\[
\left(\frac{-4}{3}\right)^7 \times \left(\frac{-4}{3}\right)^8 = \left(\frac{-4}{3}\right)^{7+8} = \left(\frac{-4}{3}\right)^{15}
\][/tex]
2. Rewrite the entire expression:
[tex]\[
\left(\frac{-4}{3}\right)^{15} \div \left(\frac{-3}{4}\right)^5
\][/tex]
3. Recognize that [tex]\(\left(\frac{-3}{4}\right)^{-1} = \left(\frac{-4}{3}\right)\)[/tex]:
[tex]\[
\left(\frac{-4}{3}\right)^{15} \div \left(\frac{-3}{4}\right)^5 = \left(\frac{-4}{3}\right)^{15} \times \left(\frac{-4}{3}\right)^5 = \left(\frac{-4}{3}\right)^{15-5} = \left(\frac{-4}{3}\right)^{10}
\][/tex]
4. Simplify for the base and exponent:
[tex]\[
\left(\frac{-4}{3}\right)^{10} = \left(\frac{-4}{3}\right)^{10}
\][/tex]
5. Calculate each part to verify intermediate steps:
- [tex]\(\left(\frac{-4}{3}\right)^7 \approx -74.83091388075754\)[/tex]
- [tex]\(\left(\frac{-3}{4}\right)^5 = \left(-0.2373046875\right)\)[/tex]
- [tex]\(\left(\frac{-4}{3}\right)^{15} \times \left(\frac{-3}{4}\right)^{-5} \approx 315.336855201217\)[/tex]
- [tex]\(\left(\frac{-4}{3}\right)^{10} \approx 17.757726633812588\)[/tex]
The numerical result is:
[tex]\[
\boxed{17.757726633812588}
\][/tex]
