Evaluate :
`sqrt((25)/(32) xx 2(13)/(18) xx 0.25)`

Answer

To evaluate the expression \( \sqrt{\frac{25}{32} \times 2 \times \frac{13}{18} \times 0.25} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{\frac{25}{32} \times 2 \times \frac{13}{18} \times 0.25} \]

### Step 2: Convert \(0.25\) to a fraction
We know that \(0.25\) can be written as \(\frac{25}{100}\). Thus, we rewrite the expression:
\[ \sqrt{\frac{25}{32} \times 2 \times \frac{13}{18} \times \frac{25}{100}} \]

### Step 3: Combine the fractions
Now, we can combine the fractions:
\[ = \sqrt{\frac{25 \times 25 \times 2 \times 13}{32 \times 100 \times 18}} \]

### Step 4: Simplify the numerator and denominator
Calculating the numerator:
\[ 25 \times 25 = 625 \]
\[ 625 \times 2 = 1250 \]
\[ 1250 \times 13 = 16250 \]

Calculating the denominator:
\[ 32 \times 100 = 3200 \]
\[ 3200 \times 18 = 57600 \]

So, we have:
\[ = \sqrt{\frac{16250}{57600}} \]

### Step 5: Simplify the fraction
We can simplify \(\frac{16250}{57600}\) by finding the greatest common divisor (GCD). The GCD of 16250 and 57600 is 50. Thus:
\[ \frac{16250 \div 50}{57600 \div 50} = \frac{325}{1152} \]

### Step 6: Rewrite the square root
Now we can rewrite the expression:
\[ = \sqrt{\frac{325}{1152}} \]

### Step 7: Separate the square root
This can be separated as:
\[ = \frac{\sqrt{325}}{\sqrt{1152}} \]

### Step 8: Simplify \(\sqrt{325}\) and \(\sqrt{1152}\)
Calculating \(\sqrt{325}\):
\[ 325 = 25 \times 13 \Rightarrow \sqrt{325} = 5\sqrt{13} \]

Calculating \(\sqrt{1152}\):
\[ 1152 = 576 \times 2 \Rightarrow \sqrt{1152} = 24\sqrt{2} \]

### Step 9: Substitute back
Now substituting back:
\[ = \frac{5\sqrt{13}}{24\sqrt{2}} \]

### Step 10: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{2}\):
\[ = \frac{5\sqrt{26}}{48} \]

### Final Answer
Thus, the final answer is:
\[ \frac{5\sqrt{26}}{48} \]

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