What is the value of the expression `(x-1)/(x^(3//4) + x^(1//2)) xx (x^(1//2) + x^(1//4))/(x^(1//2) + 1) xx x^(1//4)`, when x = 16?

Answer

To find the value of the expression

\[ \frac{x-1}{x^{3/4} + x^{1/2}} \times \frac{x^{1/2} + x^{1/4}}{x^{1/2} + 1} \times x^{1/4} \]

when \( x = 16 \), we will follow these steps:

### Step 1: Substitute the value of \( x \)
First, we substitute \( x = 16 \) into the expression.

\[ \frac{16-1}{16^{3/4} + 16^{1/2}} \times \frac{16^{1/2} + 16^{1/4}}{16^{1/2} + 1} \times 16^{1/4} \]

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

\[ 16 - 1 = 15 \]

### Step 3: Calculate \( 16^{3/4} \) and \( 16^{1/2} \)
Next, we calculate \( 16^{3/4} \) and \( 16^{1/2} \):

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

Thus,

\[ 16^{3/4} + 16^{1/2} = 8 + 4 = 12 \]

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

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

Now we can substitute these values into the expression:

\[ \frac{15}{12} \times \frac{4 + 2}{4 + 1} \times 2 \]

### Step 5: Simplify \( \frac{4 + 2}{4 + 1} \)
Calculate \( 4 + 2 \) and \( 4 + 1 \):

- \( 4 + 2 = 6 \)
- \( 4 + 1 = 5 \)

Thus,

\[ \frac{6}{5} \]

### Step 6: Substitute back into the expression
Now substitute back into the expression:

\[ \frac{15}{12} \times \frac{6}{5} \times 2 \]

### Step 7: Simplify the expression
Now we can simplify the expression step by step:

1. Multiply \( \frac{15}{12} \) and \( \frac{6}{5} \):

\[ \frac{15 \times 6}{12 \times 5} = \frac{90}{60} = \frac{3}{2} \]

2. Now multiply by \( 2 \):

\[ \frac{3}{2} \times 2 = 3 \]

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

\[ \boxed{3} \]