The value of : `sqrt((0.324xx0.081xx4.624)/(1.5625xx0.0289xx72.9xx64))` is

Options

A. `2.4`
B. `24`
C. `0.024`
D. `0.24`

Answer

To solve the expression \( \sqrt{\frac{0.324 \times 0.081 \times 4.624}{1.5625 \times 0.0289 \times 72.9 \times 64}} \), we will follow these steps:

### Step 1: Simplify the expression
First, we can rewrite the expression without decimals for easier calculation. We can express each decimal as a fraction:
- \( 0.324 = \frac{324}{1000} \)
- \( 0.081 = \frac{81}{1000} \)
- \( 4.624 = \frac{4624}{1000} \)
- \( 1.5625 = \frac{15625}{10000} \)
- \( 0.0289 = \frac{289}{10000} \)
- \( 72.9 = \frac{729}{10} \)
- \( 64 = \frac{64}{1} \)

Thus, the expression becomes:
\[ \sqrt{\frac{\frac{324 \times 81 \times 4624}{1000^3}}{\frac{15625 \times 289 \times 729 \times 64}{10000^2}}} \]

### Step 2: Simplify the fractions
Now, we can simplify the fractions:
\[ = \sqrt{\frac{324 \times 81 \times 4624 \times 10000^2}{15625 \times 289 \times 729 \times 64 \times 1000^3}} \]

### Step 3: Calculate the numerator and denominator
Now, let's calculate the values:
- **Numerator**:
- \( 324 = 18^2 \)
- \( 81 = 9^2 \)
- \( 4624 = 68^2 \) (since \( 68 \times 68 = 4624 \))

Thus, the numerator becomes:
\[ (18 \times 9 \times 68) \times 10000^2 \]

- **Denominator**:
- \( 15625 = 125^2 \)
- \( 289 = 17^2 \)
- \( 729 = 27^2 \)
- \( 64 = 8^2 \)

Thus, the denominator becomes:
\[ (125 \times 17 \times 27 \times 8) \times 1000^3 \]

### Step 4: Take the square root
Now we can take the square root of both the numerator and denominator:
\[ = \frac{18 \times 9 \times 68 \times 10000}{125 \times 17 \times 27 \times 8 \times 1000} \]

### Step 5: Cancel out common terms
Now we can cancel out \( 10000 \) and \( 1000 \):
\[ = \frac{18 \times 9 \times 68}{125 \times 17 \times 27 \times 8} \]

### Step 6: Calculate the final value
Now we can calculate the final value:
- Calculate \( 18 \times 9 = 162 \)
- Calculate \( 162 \times 68 = 11016 \)
- Calculate \( 125 \times 17 = 2125 \)
- Calculate \( 27 \times 8 = 216 \)
- Calculate \( 2125 \times 216 = 459000 \)

Thus, we have:
\[ = \frac{11016}{459000} \]

### Step 7: Simplify the fraction
Now we can simplify \( \frac{11016}{459000} \):
\[ = 0.024 \]

### Final Answer
The value of the expression is \( 0.024 \).

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