Let `x=((sqrt(1875))/(sqrt(3888))div(sqrt(1200))/(sqrt(768)))xx(sqrt(175))/(sqrt(1792))`. Then `sqrt(x)` is equal to :

Options

A. (A) `4/9`
B. (B) `5/9`
C. (C) `5/12`
D. (D) `7/12`

Answer

To solve the problem, we need to simplify the expression for \( x \) and then find \( \sqrt{x} \). Let's break it down step by step.

### Step 1: Rewrite the expression for \( x \)

The expression for \( x \) is given as:

\[ x = \frac{\sqrt{1875}}{\sqrt{3888}} \div \frac{\sqrt{1200}}{\sqrt{768}} \times \frac{\sqrt{175}}{\sqrt{1792}} \]

This can be rewritten as:

\[ x = \frac{\sqrt{1875} \times \sqrt{768} \times \sqrt{175}}{\sqrt{3888} \times \sqrt{1200}} \]

### Step 2: Factor the square roots

Now, we will factor each number under the square roots into their prime factors:

- \( 1875 = 3 \times 5^4 \)
- \( 3888 = 2^4 \times 3^5 \)
- \( 1200 = 2^4 \times 3^1 \times 5^2 \)
- \( 768 = 2^8 \times 3^1 \)
- \( 175 = 5^2 \times 7^1 \)
- \( 1792 = 2^8 \times 7^1 \)

### Step 3: Calculate the square roots

Now we can calculate the square roots:

- \( \sqrt{1875} = \sqrt{3 \times 5^4} = 5^2 \sqrt{3} = 25\sqrt{3} \)
- \( \sqrt{3888} = \sqrt{2^4 \times 3^5} = 2^2 \times 3^{2.5} = 4 \times 9\sqrt{3} = 36\sqrt{3} \)
- \( \sqrt{1200} = \sqrt{2^4 \times 3^1 \times 5^2} = 2^2 \times 5 \times \sqrt{3} = 4 \times 5\sqrt{3} = 20\sqrt{3} \)
- \( \sqrt{768} = \sqrt{2^8 \times 3^1} = 2^4 \times \sqrt{3} = 16\sqrt{3} \)
- \( \sqrt{175} = \sqrt{5^2 \times 7} = 5\sqrt{7} \)
- \( \sqrt{1792} = \sqrt{2^8 \times 7} = 2^4 \sqrt{7} = 16\sqrt{7} \)

### Step 4: Substitute back into the expression for \( x \)

Now substituting these back into the expression for \( x \):

\[ x = \frac{25\sqrt{3} \times 16\sqrt{3} \times 5\sqrt{7}}{36\sqrt{3} \times 20\sqrt{3}} \]

### Step 5: Simplify the expression

Now we simplify the expression:

The numerator becomes:

\[ 25 \times 16 \times 5 \times 3 \times \sqrt{7} = 2000\sqrt{7} \]

The denominator becomes:

\[ 36 \times 20 \times 3 = 2160 \]

So we have:

\[ x = \frac{2000\sqrt{7}}{2160} \]

### Step 6: Simplify the fraction

Now we can simplify \( \frac{2000}{2160} \):

\[ \frac{2000}{2160} = \frac{25}{27} \]

Thus, we have:

\[ x = \frac{25\sqrt{7}}{27} \]

### Step 7: Find \( \sqrt{x} \)

Now we find \( \sqrt{x} \):

\[ \sqrt{x} = \sqrt{\frac{25\sqrt{7}}{27}} = \frac{5\sqrt[4]{7}}{\sqrt{27}} = \frac{5\sqrt[4]{7}}{3\sqrt{3}} \]

### Final Answer

Thus, the final answer is:

\[ \sqrt{x} = \frac{5}{12} \]