Evaluate : `(4)/((216)^(-2//3))+(1)/((256)^(-3//4))+(2)/((343)^(-1//3))`

Answer

To evaluate the expression

\[ \frac{4}{(216)^{-\frac{2}{3}}} + \frac{1}{(256)^{-\frac{3}{4}}} + \frac{2}{(343)^{-\frac{1}{3}}} \]

we will follow these steps:

### Step 1: Rewrite the bases in terms of their prime factors.

1. **Rewrite 216, 256, and 343:**
- \( 216 = 6^3 \)
- \( 256 = 4^4 \)
- \( 343 = 7^3 \)

So, we can rewrite the expression as:

\[ \frac{4}{(6^3)^{-\frac{2}{3}}} + \frac{1}{(4^4)^{-\frac{3}{4}}} + \frac{2}{(7^3)^{-\frac{1}{3}}} \]

### Step 2: Apply the power of a power rule.

2. **Use the power of a power rule:**
- \((a^m)^n = a^{m \cdot n}\)

Applying this rule:

\[ \frac{4}{6^{-2}} + \frac{1}{4^{-3}} + \frac{2}{7^{-1}} \]

### Step 3: Simplify the negative exponents.

3. **Convert negative exponents to positive:**
- \(\frac{1}{a^{-n}} = a^n\)

Thus, we have:

\[ 4 \cdot 6^2 + 1 \cdot 4^3 + 2 \cdot 7^1 \]

### Step 4: Calculate the powers.

4. **Calculate the powers:**
- \(6^2 = 36\)
- \(4^3 = 64\)
- \(7^1 = 7\)

Now, substituting these values:

\[ 4 \cdot 36 + 1 \cdot 64 + 2 \cdot 7 \]

### Step 5: Multiply and add the results.

5. **Perform the multiplications:**
- \(4 \cdot 36 = 144\)
- \(1 \cdot 64 = 64\)
- \(2 \cdot 7 = 14\)

Now, add these results:

\[ 144 + 64 + 14 \]

### Step 6: Final addition.

6. **Add the results together:**
- \(144 + 64 = 208\)
- \(208 + 14 = 222\)

Thus, the final answer is:

\[ \boxed{222} \]