Evaluate :
` [5(8^((1)/(3))+ 27 ^((1)/(3))) ^(3) ]^((1)/(4))`

Answer

To evaluate the expression \( \left[ 5(8^{\frac{1}{3}} + 27^{\frac{1}{3}})^{3} \right]^{\frac{1}{4}} \), we can follow these steps:

### Step 1: Simplify the cube roots
First, we simplify \( 8^{\frac{1}{3}} \) and \( 27^{\frac{1}{3}} \):
- \( 8 = 2^3 \) so \( 8^{\frac{1}{3}} = (2^3)^{\frac{1}{3}} = 2^{1} = 2 \)
- \( 27 = 3^3 \) so \( 27^{\frac{1}{3}} = (3^3)^{\frac{1}{3}} = 3^{1} = 3 \)

### Step 2: Substitute back into the expression
Now substitute these values back into the expression:
\[ 5(8^{\frac{1}{3}} + 27^{\frac{1}{3}}) = 5(2 + 3) = 5 \times 5 = 25 \]

### Step 3: Cube the result
Next, we cube the result:
\[ (25)^{3} = 15625 \]

### Step 4: Take the fourth root
Finally, we take the fourth root of \( 15625 \):
\[ \left(15625\right)^{\frac{1}{4}} \]

To find \( \left(15625\right)^{\frac{1}{4}} \), we can recognize that:
\[ 15625 = 25^3 = (5^2)^3 = 5^6 \]
Thus,
\[ \left(15625\right)^{\frac{1}{4}} = (5^6)^{\frac{1}{4}} = 5^{\frac{6}{4}} = 5^{\frac{3}{2}} = 5^{1.5} = 5 \sqrt{5} \]

### Final Answer
So, the final answer is:
\[ 5 \sqrt{5} \]