Compute ` log_(ab)(root(3)a//sqrtb)" if " log_(ab) a = 4`.

Explanation

To compute \( \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right) \) given that \( \log_{ab} a = 4 \), we will follow these steps:

### Step 1: Rewrite the logarithm using the change of base formula
Using the change of base formula, we can express \( \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right) \) as:
\[ \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right) = \frac{\log_a\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right)}{\log_a(ab)} \]

### Step 2: Simplify the logarithm in the numerator
Now, we can simplify the numerator:
\[ \log_a\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right) = \log_a\left(\sqrt[3]{a}\right) - \log_a\left(\sqrt{b}\right) \]
Using the property \( \log_a(m^n) = n \cdot \log_a(m) \):
\[ = \frac{1}{3} \log_a(a) - \frac{1}{2} \log_a(b) \]
Since \( \log_a(a) = 1 \):
\[ = \frac{1}{3} - \frac{1}{2} \log_a(b) \]

### Step 3: Simplify the denominator
Next, we simplify the denominator:
\[ \log_a(ab) = \log_a(a) + \log_a(b) = 1 + \log_a(b) \]

### Step 4: Substitute back into the logarithm expression
Now substituting back into our expression:
\[ \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right) = \frac{\frac{1}{3} - \frac{1}{2} \log_a(b)}{1 + \log_a(b)} \]

### Step 5: Use the given information
We know that \( \log_{ab} a = 4 \). Using the change of base formula again:
\[ \log_{ab} a = \frac{\log_a a}{\log_a(ab)} = \frac{1}{1 + \log_a(b)} = 4 \]
This implies:
\[ 1 = 4(1 + \log_a(b)) \implies 1 = 4 + 4\log_a(b) \implies 4\log_a(b) = 1 - 4 = -3 \implies \log_a(b) = -\frac{3}{4} \]

### Step 6: Substitute \( \log_a(b) \) into the expression
Now substituting \( \log_a(b) = -\frac{3}{4} \) into our expression:
\[ \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right) = \frac{\frac{1}{3} - \frac{1}{2}\left(-\frac{3}{4}\right)}{1 + \left(-\frac{3}{4}\right)} \]
Calculating the numerator:
\[ = \frac{\frac{1}{3} + \frac{3}{8}}{1 - \frac{3}{4}} = \frac{\frac{1}{3} + \frac{3}{8}}{\frac{1}{4}} \]

### Step 7: Find a common denominator for the numerator
Finding a common denominator for \( \frac{1}{3} \) and \( \frac{3}{8} \):
\[ \frac{1}{3} = \frac{8}{24}, \quad \frac{3}{8} = \frac{9}{24} \implies \frac{8 + 9}{24} = \frac{17}{24} \]
Thus, the expression becomes:
\[ \frac{\frac{17}{24}}{\frac{1}{4}} = \frac{17}{24} \cdot 4 = \frac{17}{6} \]

### Final Answer
Therefore, the value of \( \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right) \) is:
\[ \boxed{\frac{17}{6}} \]