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Let's evaluate the expression:

\[
\frac{4}{(216)^{-\frac{2}{3}}} + \frac{1}{(256)^{\frac{3}{4}}} + \frac{2}{(243)^{-\frac{1}{5}}}
\]

We'll evaluate each term separately.

### Step 1: Evaluate \(\frac{4}{(216)^{-\frac{2}{3}}}\)

1. First, rewrite the negative exponent:
   \[
   (216)^{-\frac{2}{3}} = \frac{1}{(216)^{\frac{2}{3}}}
   \]
   So, \(\frac{4}{(216)^{-\frac{2}{3}}} = 4 \times (216)^{\frac{2}{3}}\).

2. Now calculate \((216)^{\frac{2}{3}}\):
   - 216 can be written as \(6^3\).
   - Thus, \((216)^{\frac{2}{3}} = (6^3)^{\frac{2}{3}} = 6^2 = 36\).

3. Therefore:
   \[
   \frac{4}{(216)^{-\frac{2}{3}}} = 4 \times 36 = 144
   \]

### Step 2: Evaluate \(\frac{1}{(256)^{\frac{3}{4}}}\)

1. Rewrite \(256\) as \(4^4\).
2. Then \((256)^{\frac{3}{4}} = (4^4)^{\frac{3}{4}} = 4^3 = 64\).
3. Therefore:
   \[
   \frac{1}{(256)^{\frac{3}{4}}} = \frac{1}{64}
   \]

### Step 3: Evaluate \(\frac{2}{(243)^{-\frac{1}{5}}}\)

1. Rewrite the negative exponent:
   \[
   (243)^{-\frac{1}{5}} = \frac{1}{(243)^{\frac{1}{5}}}
   \]
   So, \(\frac{2}{(243)^{-\frac{1}{5}}} = 2 \times (243)^{\frac{1}{5}}\).

2. Now calculate \((243)^{\frac{1}{5}}\):
   - 243 can be written as \(3^5\).
   - Thus, \((243)^{\frac{1}{5}} = (3^5)^{\frac{1}{5}} = 3\).

3. Therefore:
   \[
   \frac{2}{(243)^{-\frac{1}{5}}} = 2 \times 3 = 6
   \]

### Step 4: Add the Results

Now combine all terms:
\[
144 + \frac{1}{64} + 6
\]

Convert \(144\) and \(6\) to a common denominator of 64 to add the terms precisely.

1. \(144 = \frac{9216}{64}\)
2. \(6 = \frac{384}{64}\)
3. Therefore:
   \[
   144 + \frac{1}{64} + 6 = \frac{9216 + 1 + 384}{64} = \frac{9601}{64}
   \]

So, the final result is:
\[
\frac{9601}{64} \quad \text{or approximately } 150.015625
\]

Would you like more details, or have any questions?

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Here are five related questions to explore:

1. How do fractional exponents relate to roots in expressions?
2. What are the rules for handling negative exponents in a fraction?
3. How can we simplify complex exponents with prime factorization?
4. How do fractional exponents and radical forms relate in algebra?
5. How would the solution change if the terms had different bases?

**Tip:** Breaking down large exponents into simpler prime factors can often make calculations easier and quicker.

Find the value of \( \frac{4}{(216)^{-\frac{2}{3}}} + \frac{1}{(256)^{\frac{3}{4}}} + \frac{2}{(243)^{-\frac{1}{5}}} \)

Learn how to evaluate the expression \( \frac{4}{(216)^{-\frac{2}{3}}} + \frac{1}{(256)^{\frac{3}{4}}} + \frac{2}{(243)^{-\frac{1}{5}}} \) by simplifying fractional and negative exponents. This solution covers exponentiation rules and key algebraic concepts for high school students in Grades 9-11.

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