If `x= (sqrt126 xx sqrt63 xx sqrt45)/(sqrt147 xx sqrt243)`, then the value of x is

Options

A. `sqrt5`
B. `sqrt10`
C. 10
D. 5

Answer

To solve the expression \( x = \frac{\sqrt{126} \times \sqrt{63} \times \sqrt{45}}{\sqrt{147} \times \sqrt{243}} \), we will simplify each square root by prime factorization and then combine the results.

### Step 1: Prime Factorization of Each Term
1. **For \( \sqrt{126} \)**:
- \( 126 = 2 \times 3^2 \times 7 \)
- So, \( \sqrt{126} = \sqrt{2 \times 3^2 \times 7} = 3\sqrt{14} \)

2. **For \( \sqrt{63} \)**:
- \( 63 = 3^2 \times 7 \)
- So, \( \sqrt{63} = \sqrt{3^2 \times 7} = 3\sqrt{7} \)

3. **For \( \sqrt{45} \)**:
- \( 45 = 3^2 \times 5 \)
- So, \( \sqrt{45} = \sqrt{3^2 \times 5} = 3\sqrt{5} \)

4. **For \( \sqrt{147} \)**:
- \( 147 = 3 \times 7^2 \)
- So, \( \sqrt{147} = \sqrt{3 \times 7^2} = 7\sqrt{3} \)

5. **For \( \sqrt{243} \)**:
- \( 243 = 3^5 \)
- So, \( \sqrt{243} = \sqrt{3^5} = 9\sqrt{3} \)

### Step 2: Substitute the Simplified Square Roots into the Expression
Now substituting these back into the expression for \( x \):

\[ x = \frac{(3\sqrt{14}) \times (3\sqrt{7}) \times (3\sqrt{5})}{(7\sqrt{3}) \times (9\sqrt{3})} \]

### Step 3: Simplify the Numerator and Denominator
1. **Numerator**:
- \( 3\sqrt{14} \times 3\sqrt{7} \times 3\sqrt{5} = 27\sqrt{14 \times 7 \times 5} = 27\sqrt{490} \)

2. **Denominator**:
- \( (7\sqrt{3}) \times (9\sqrt{3}) = 63 \times 3 = 189 \)

### Step 4: Combine the Results
Thus, we have:

\[ x = \frac{27\sqrt{490}}{189} \]

### Step 5: Simplify the Fraction
Now simplify \( \frac{27}{189} \):

\[ \frac{27}{189} = \frac{1}{7} \]

So, we can write:

\[ x = \frac{1}{7} \sqrt{490} \]

### Step 6: Simplify \( \sqrt{490} \)
Now, simplify \( \sqrt{490} \):

- \( 490 = 2 \times 5 \times 7^2 \)
- So, \( \sqrt{490} = 7\sqrt{10} \)

### Final Result
Thus, substituting back, we get:

\[ x = \frac{1}{7} \times 7\sqrt{10} = \sqrt{10} \]

### Conclusion
The value of \( x \) is \( \sqrt{10} \).

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