What approximate value will come in place of (?) in the following questions ?
`(5.01)^(3) xx (23.99)^(2) div (44.98)^(3) = (?)^(2)`

Options

A. `(13)/(9)`
B. `(8)/(9)`
C. `(5)/(7)`
D. `(12)/(7)`

Answer

To solve the problem `(5.01)^(3) * (23.99)^(2) / (44.98)^(3) = (?)^(2)`, we will approximate the numbers and simplify step by step.

### Step 1: Approximate the numbers
We will round the numbers to make calculations easier:
- \(5.01 \approx 5\)
- \(23.99 \approx 24\)
- \(44.98 \approx 45\)

Thus, we can rewrite the equation as:
\[ (5)^{3} * (24)^{2} / (45)^{3} = (?)^{2} \]

### Step 2: Calculate the powers
Now, we will calculate the powers:
- \(5^{3} = 125\)
- \(24^{2} = 576\)
- \(45^{3} = 45 * 45 * 45\)

Calculating \(45^{3}\):
\[ 45 * 45 = 2025 \]
\[ 2025 * 45 = 91125 \]

So, we have:
\[ (5)^{3} * (24)^{2} = 125 * 576 \]

### Step 3: Calculate \(125 * 576\)
Now we will calculate \(125 * 576\):
\[ 125 * 576 = 72000 \]

### Step 4: Substitute back into the equation
Now we substitute back into the equation:
\[ \frac{72000}{91125} = (?)^{2} \]

### Step 5: Simplify the fraction
Next, we simplify the fraction:
\[ \frac{72000}{91125} \]

To simplify, we can find the GCD or use a calculator to get the approximate value. After simplification, we find:
\[ \frac{72000}{91125} \approx 0.79 \]

### Step 6: Find the square root
Now we take the square root of both sides:
\[ ? = \sqrt{0.79} \approx 0.89 \]

### Step 7: Compare with options
Now we compare this value with the options given:
- \( \frac{8}{9} \approx 0.89 \)
- Other options are smaller or larger.

Thus, the approximate value of \( ? \) is:
\[ ? \approx \frac{8}{9} \]

### Final Answer
The approximate value that comes in place of (?) is:
\[ \frac{8}{9} \]