To solve the given expression:

$\left( \frac{1}{125} \right)^{-\frac{2}{3}} + (8)^{\frac{4}{3}} + (256)^{\frac{1}{4}}$

Let's break down and solve each part separately.

1. Calculating $\left( \frac{1}{125} \right)^{-\frac{2}{3}}$

$\left( \frac{1}{125} \right)^{-\frac{2}{3}} = 125^{\frac{2}{3}}$

Since $125 = 5^3$, we have: $125^{\frac{2}{3}} = (5^3)^{\frac{2}{3}} = 5^{2} = 25$

2. Calculating $(8)^{\frac{4}{3}}$

$(8)^{\frac{4}{3}} = (2^3)^{\frac{4}{3}} = 2^{4} = 16$

3. Calculating $(256)^{\frac{1}{4}}$

$(256)^{\frac{1}{4}} = (2^8)^{\frac{1}{4}} = 2^{2} = 4$

4. Adding the results

$25 + 16 + 4 = 45$

Thus, the final result is:

$\boxed{45}$

If you have any questions or need further details, feel free to ask!

Related Questions:

1. How do you simplify expressions with fractional exponents?
2. Can you explain the property of exponents used in these calculations?
3. How do you solve expressions involving negative exponents?
4. What is the significance of prime factorization in simplifying roots?
5. How do you handle nested exponents?

Tip:

Always remember to express numbers in their prime factorization form when dealing with roots and exponents. It simplifies calculations and helps in understanding the problem better.