The expression `a^((2)/(3)){a^((1)/(3))(a^((1)/(4)))^(4)}^((1)/(4))` is equal to

Options

A. `a^((1)/(2))`
B. `a^((1)/(6))`
C. a
D. 1

Answer

To solve the expression \( a^{\frac{2}{3}} \left( a^{\frac{1}{3}} \left( a^{\frac{1}{4}} \right)^{4} \right)^{\frac{1}{4}} \), we will follow these steps:

### Step 1: Simplify the inner expression
First, we simplify the inner expression \( \left( a^{\frac{1}{4}} \right)^{4} \).

Using the power of a power property, we multiply the exponents:
\[ \left( a^{\frac{1}{4}} \right)^{4} = a^{\frac{1}{4} \times 4} = a^{1} \]

### Step 2: Substitute back into the expression
Now substitute \( a^{1} \) back into the expression:
\[ a^{\frac{2}{3}} \left( a^{\frac{1}{3}} \cdot a^{1} \right)^{\frac{1}{4}} \]

### Step 3: Combine the terms inside the parentheses
Next, we combine the terms inside the parentheses:
\[ a^{\frac{1}{3}} \cdot a^{1} = a^{\frac{1}{3} + 1} = a^{\frac{1}{3} + \frac{3}{3}} = a^{\frac{4}{3}} \]

### Step 4: Substitute back into the expression
Now substitute back:
\[ a^{\frac{2}{3}} \left( a^{\frac{4}{3}} \right)^{\frac{1}{4}} \]

### Step 5: Simplify the exponent
Now simplify \( \left( a^{\frac{4}{3}} \right)^{\frac{1}{4}} \):
\[ \left( a^{\frac{4}{3}} \right)^{\frac{1}{4}} = a^{\frac{4}{3} \cdot \frac{1}{4}} = a^{\frac{4}{12}} = a^{\frac{1}{3}} \]

### Step 6: Combine the terms
Now we have:
\[ a^{\frac{2}{3}} \cdot a^{\frac{1}{3}} = a^{\frac{2}{3} + \frac{1}{3}} = a^{\frac{3}{3}} = a^{1} \]

### Final Answer
Thus, the expression simplifies to:
\[ \boxed{a} \]