What approximate value should come in place of Question mark (?) in the following equation ?
`sqrt(82 div 8.76 div 9.25 xx 3.76) + (181)/(91.10) xx (473.92)/(237.40) + 7.91 = ?^((1)/(2))`

Options

A. 100
B. 400
C. 256
D. 196

Answer

To solve the equation step by step, we need to simplify the expression given in the question. The equation is:

\[ \sqrt{\frac{82}{8.76 \div 9.25 \times 3.76}} + \frac{181}{91.10} \times \frac{473.92}{237.40} + 7.91 = ?^{\frac{1}{2}} \]

### Step 1: Simplify the first term inside the square root

We start with the first term:

\[ \frac{82}{8.76 \div 9.25 \times 3.76} \]

First, we need to calculate \(8.76 \div 9.25\):

\[ 8.76 \div 9.25 \approx 0.946 \]

Now, multiply this result by \(3.76\):

\[ 0.946 \times 3.76 \approx 3.56 \]

Now we can substitute this back into our expression:

\[ \frac{82}{3.56} \approx 23.0 \]

### Step 2: Calculate the square root

Now we take the square root of the result:

\[ \sqrt{23.0} \approx 4.8 \]

### Step 3: Simplify the second term

Next, we simplify the second term:

\[ \frac{181}{91.10} \times \frac{473.92}{237.40} \]

Calculating \( \frac{181}{91.10} \):

\[ \frac{181}{91.10} \approx 1.99 \]

Now, calculate \( \frac{473.92}{237.40} \):

\[ \frac{473.92}{237.40} \approx 1.99 \]

Now multiply these two results together:

\[ 1.99 \times 1.99 \approx 3.96 \]

### Step 4: Add the constant term

Now we add \(7.91\) to the result from the second term:

\[ 4.8 + 3.96 + 7.91 \approx 16.67 \]

### Step 5: Set the equation

Now we have:

\[ \sqrt{?} = 16.67 \]

### Step 6: Square both sides to find the value of ?

Squaring both sides gives us:

\[ ? = (16.67)^2 \approx 278.0889 \]

### Step 7: Choose the closest approximate value

Among the options provided (100, 400, 256, 296), the closest approximate value to \(278.0889\) is \(296\).

Thus, the final answer is:

\[ ? \approx 296 \]