The value of `(sqrt(72)xxsqrt(363)xxsqrt(175))/(sqrt(32)xxsqrt(147)xxsqrt(252))` is

Options

A. A) `(55)/(42)`
B. B) `(45)/(56)`
C. C) `(45)/(28)`
D. D) `(55)/(28)`

Answer

To solve the expression \((\sqrt{72} \times \sqrt{363} \times \sqrt{175}) / (\sqrt{32} \times \sqrt{147} \times \sqrt{252})\), we can follow these steps:

### Step 1: Combine the square roots
We can combine the square roots in the numerator and denominator:
\[ \frac{\sqrt{72 \times 363 \times 175}}{\sqrt{32 \times 147 \times 252}} \]

### Step 2: Simplify the expression under the square roots
Now we will calculate the products in the numerator and denominator:
- Numerator: \(72 \times 363 \times 175\)
- Denominator: \(32 \times 147 \times 252\)

### Step 3: Factor the numbers
Let's factor each number into its prime factors:
- \(72 = 2^3 \times 3^2\)
- \(363 = 3 \times 11^2\)
- \(175 = 5^2 \times 7\)
- \(32 = 2^5\)
- \(147 = 3 \times 7^2\)
- \(252 = 2^2 \times 3^2 \times 7\)

### Step 4: Substitute the factors into the products
Now substitute the factors back into the products:
\[ 72 \times 363 \times 175 = (2^3 \times 3^2) \times (3 \times 11^2) \times (5^2 \times 7) \]
\[ 32 \times 147 \times 252 = (2^5) \times (3 \times 7^2) \times (2^2 \times 3^2 \times 7) \]

### Step 5: Combine the factors
Combine the factors:
Numerator:
\[ = 2^3 \times 3^3 \times 11^2 \times 5^2 \times 7 \]
Denominator:
\[ = 2^{5+2} \times 3^{1+2} \times 7^{1+2} = 2^7 \times 3^3 \times 7^3 \]

### Step 6: Simplify the fraction
Now we can simplify the fraction:
\[ \frac{2^3 \times 3^3 \times 11^2 \times 5^2 \times 7}{2^7 \times 3^3 \times 7^3} \]
This simplifies to:
\[ \frac{11^2 \times 5^2}{2^{7-3} \times 7^{3-1}} = \frac{11^2 \times 5^2}{2^4 \times 7^2} \]

### Step 7: Take the square root
Now we take the square root of the simplified fraction:
\[ \sqrt{\frac{11^2 \times 5^2}{2^4 \times 7^2}} = \frac{11 \times 5}{2^2 \times 7} = \frac{55}{4 \times 7} = \frac{55}{28} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{55}{28} \]