The square root of:
`(2(1/3) -1(1/6))/(2(1/3) + 1(1/6))` is:

Options

A. `1/9`
B. `1/sqrt(3)`
C. `1/(3sqrt(3))`
D. 3

Answer

To solve the expression \( \sqrt{\frac{2\frac{1}{3} - 1\frac{1}{6}}{2\frac{1}{3} + 1\frac{1}{6}}} \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers \( 2\frac{1}{3} \) and \( 1\frac{1}{6} \) into improper fractions.

- For \( 2\frac{1}{3} \):
\[ 2\frac{1}{3} = 2 \times 3 + 1 = 6 + 1 = \frac{7}{3} \]

- For \( 1\frac{1}{6} \):
\[ 1\frac{1}{6} = 1 \times 6 + 1 = 6 + 1 = \frac{7}{6} \]

### Step 2: Substitute the Improper Fractions into the Expression
Now substitute these values back into the expression:
\[ \sqrt{\frac{\frac{7}{3} - \frac{7}{6}}{\frac{7}{3} + \frac{7}{6}}} \]

### Step 3: Find a Common Denominator
To perform the subtraction and addition in the numerator and denominator, we need a common denominator. The least common multiple of 3 and 6 is 6.

- For the numerator:
\[ \frac{7}{3} = \frac{7 \times 2}{3 \times 2} = \frac{14}{6} \]
So,
\[ \frac{7}{3} - \frac{7}{6} = \frac{14}{6} - \frac{7}{6} = \frac{14 - 7}{6} = \frac{7}{6} \]

- For the denominator:
\[ \frac{7}{3} + \frac{7}{6} = \frac{14}{6} + \frac{7}{6} = \frac{14 + 7}{6} = \frac{21}{6} \]

### Step 4: Substitute Back into the Expression
Now substitute the results back into the expression:
\[ \sqrt{\frac{\frac{7}{6}}{\frac{21}{6}}} \]

### Step 5: Simplify the Fraction
When dividing fractions, we multiply by the reciprocal:
\[ \frac{\frac{7}{6}}{\frac{21}{6}} = \frac{7}{6} \times \frac{6}{21} = \frac{7}{21} = \frac{1}{3} \]

### Step 6: Take the Square Root
Now we take the square root of the simplified expression:
\[ \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \]

### Final Answer
Thus, the value of the original expression is:
\[ \frac{1}{\sqrt{3}} \]