Evaluate :
` (1)/( ( 216)^((-2)/(3)) ) + (1)/((27) ^((-4)/(3))) `

Answer

To evaluate the expression \( \frac{1}{(216)^{-\frac{2}{3}}} + \frac{1}{(27)^{-\frac{4}{3}}} \), we will follow these steps:

### Step 1: Rewrite the expression
We can rewrite the expression as:
\[ \frac{1}{216^{-\frac{2}{3}}} + \frac{1}{27^{-\frac{4}{3}}} \]

### Step 2: Apply the property of exponents
Using the property \( a^{-m} = \frac{1}{a^m} \), we can rewrite the expression:
\[ 216^{-\frac{2}{3}} = \frac{1}{216^{\frac{2}{3}}} \quad \text{and} \quad 27^{-\frac{4}{3}} = \frac{1}{27^{\frac{4}{3}}} \]
Thus, the expression becomes:
\[ 216^{\frac{2}{3}} + 27^{\frac{4}{3}} \]

### Step 3: Simplify \( 216^{\frac{2}{3}} \)
We know that \( 216 = 6^3 \), so we can write:
\[ 216^{\frac{2}{3}} = (6^3)^{\frac{2}{3}} = 6^{3 \cdot \frac{2}{3}} = 6^2 = 36 \]

### Step 4: Simplify \( 27^{\frac{4}{3}} \)
Similarly, since \( 27 = 3^3 \), we have:
\[ 27^{\frac{4}{3}} = (3^3)^{\frac{4}{3}} = 3^{3 \cdot \frac{4}{3}} = 3^4 = 81 \]

### Step 5: Add the results
Now we can add the two results:
\[ 36 + 81 = 117 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{117} \]

---