`(7)1/2-[(2)1/4div{(1)1/4-1/2((1)1/2-1/3-1/6)}]`is equal to

Options

A. (2/9)
B. 1
C. (4)`1/2`
D. (1)77/288

Answer

To solve the expression \( (7)^{1/2} - \left[ (2)^{1/4} \div \left\{ (1)^{1/4} - \frac{1}{2} \left( (1)^{1/2} - \frac{1}{3} - \frac{1}{6} \right) \right\} \right] \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert the mixed numbers into improper fractions:
- \( 7 \frac{1}{2} = \frac{15}{2} \)
- \( 2 \frac{1}{4} = \frac{9}{4} \)
- \( 1 \frac{1}{4} = \frac{5}{4} \)

### Step 2: Rewrite the Expression
The expression now looks like:
\[ \frac{15}{2} - \left[ \frac{9}{4} \div \left\{ \frac{5}{4} - \frac{1}{2} \left( (1)^{1/2} - \frac{1}{3} - \frac{1}{6} \right) \right\} \right] \]

### Step 3: Solve the Inner Bracket
Now, we simplify the inner bracket:
\[ (1)^{1/2} - \frac{1}{3} - \frac{1}{6} \]
Calculating \( (1)^{1/2} = 1 \):
\[ 1 - \frac{1}{3} - \frac{1}{6} \]
Finding a common denominator (which is 6):
\[ 1 = \frac{6}{6}, \quad \frac{1}{3} = \frac{2}{6}, \quad \frac{1}{6} = \frac{1}{6} \]
Thus:
\[ 1 - \frac{1}{3} - \frac{1}{6} = \frac{6}{6} - \frac{2}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \]

### Step 4: Substitute Back into the Expression
Now substitute back into the expression:
\[ \frac{15}{2} - \left[ \frac{9}{4} \div \left( \frac{5}{4} - \frac{1}{2} \cdot \frac{1}{2} \right) \right] \]
Calculating \( \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \):
\[ \frac{5}{4} - \frac{1}{4} = \frac{4}{4} = 1 \]

### Step 5: Continue Simplifying
Now we have:
\[ \frac{15}{2} - \left[ \frac{9}{4} \div 1 \right] = \frac{15}{2} - \frac{9}{4} \]

### Step 6: Find a Common Denominator
To subtract these fractions, we need a common denominator. The least common multiple of 2 and 4 is 4:
\[ \frac{15}{2} = \frac{30}{4} \]
Now we can subtract:
\[ \frac{30}{4} - \frac{9}{4} = \frac{30 - 9}{4} = \frac{21}{4} \]

### Step 7: Convert Back to Mixed Number
Finally, convert \( \frac{21}{4} \) back to a mixed number:
\[ \frac{21}{4} = 5 \frac{1}{4} \]

### Final Answer
Thus, the final answer is:
\[ 5 \frac{1}{4} \]