If `x = 16`, then find the value of the expression `(x-1)/( x^(3//4) + x^(1//2) ) . ( x^(1//2) + x^(1//4) )/( x^(1//2) +1) . x^(1//4)`.

Options

A. 9
B. 3
C. 27
D. `3^0`

Answer

To solve the expression \((x-1)/( x^{3/4} + x^{1/2} ) \cdot ( x^{1/2} + x^{1/4})/( x^{1/2} +1) \cdot x^{1/4}\) for \(x = 16\), we will follow these steps:

### Step 1: Substitute the value of \(x\)
We start by substituting \(x = 16\) into the expression.

\[ \text{Expression} = \frac{(16-1)}{(16^{3/4} + 16^{1/2})} \cdot \frac{(16^{1/2} + 16^{1/4})}{(16^{1/2} + 1)} \cdot 16^{1/4} \]

### Step 2: Simplify the numerator
Calculate \(16 - 1\):

\[ 16 - 1 = 15 \]

### Step 3: Calculate the powers of \(16\)
Now, calculate \(16^{3/4}\), \(16^{1/2}\), and \(16^{1/4}\):

- \(16^{1/2} = \sqrt{16} = 4\)
- \(16^{1/4} = \sqrt[4]{16} = 2\)
- \(16^{3/4} = (16^{1/4})^3 = 2^3 = 8\)

### Step 4: Substitute these values back into the expression
Now substitute these values back into the expression:

\[ \text{Expression} = \frac{15}{(8 + 4)} \cdot \frac{(4 + 2)}{(4 + 1)} \cdot 2 \]

### Step 5: Simplify the denominator
Calculate \(8 + 4\) and \(4 + 1\):

\[ 8 + 4 = 12 \]
\[ 4 + 1 = 5 \]

### Step 6: Substitute and simplify further
Now substitute these values back into the expression:

\[ \text{Expression} = \frac{15}{12} \cdot \frac{6}{5} \cdot 2 \]

### Step 7: Simplify \(\frac{15}{12}\)
Simplify \(\frac{15}{12}\):

\[ \frac{15}{12} = \frac{5}{4} \]

### Step 8: Substitute and simplify
Now substitute this back into the expression:

\[ \text{Expression} = \frac{5}{4} \cdot \frac{6}{5} \cdot 2 \]

### Step 9: Cancel out the \(5\)
The \(5\) in the numerator and denominator cancels out:

\[ \text{Expression} = \frac{6}{4} \cdot 2 \]

### Step 10: Simplify \(\frac{6}{4}\)
Simplify \(\frac{6}{4}\):

\[ \frac{6}{4} = \frac{3}{2} \]

### Step 11: Final multiplication
Now multiply \(\frac{3}{2} \cdot 2\):

\[ \text{Expression} = 3 \]

### Final Answer
Thus, the value of the expression when \(x = 16\) is:

\[ \boxed{3} \]