IF `(sqrt126xxsqrt63xxsqrt45)/(sqrt147xxsqrt243)` then the value of x is

Options

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

Answer

To solve the expression \(\frac{\sqrt{126} \times \sqrt{63} \times \sqrt{45}}{\sqrt{147} \times \sqrt{243}}\) and find the value of \(x\), we will simplify each square root step by step.

### Step 1: Simplify each square root

1. **For \(\sqrt{126}\)**:
\[ 126 = 9 \times 14 = 3^2 \times 14 \implies \sqrt{126} = \sqrt{9 \times 14} = \sqrt{9} \times \sqrt{14} = 3\sqrt{14} \]

2. **For \(\sqrt{63}\)**:
\[ 63 = 9 \times 7 = 3^2 \times 7 \implies \sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7} \]

3. **For \(\sqrt{45}\)**:
\[ 45 = 9 \times 5 = 3^2 \times 5 \implies \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \]

4. **For \(\sqrt{147}\)**:
\[ 147 = 49 \times 3 = 7^2 \times 3 \implies \sqrt{147} = \sqrt{49 \times 3} = \sqrt{49} \times \sqrt{3} = 7\sqrt{3} \]

5. **For \(\sqrt{243}\)**:
\[ 243 = 81 \times 3 = 9^2 \times 3 \implies \sqrt{243} = \sqrt{81 \times 3} = \sqrt{81} \times \sqrt{3} = 9\sqrt{3} \]

### Step 2: Substitute back into the expression

Now we can substitute these simplified forms back into the original expression:
\[ \frac{\sqrt{126} \times \sqrt{63} \times \sqrt{45}}{\sqrt{147} \times \sqrt{243}} = \frac{(3\sqrt{14}) \times (3\sqrt{7}) \times (3\sqrt{5})}{(7\sqrt{3}) \times (9\sqrt{3})} \]

### 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 and simplify the fraction

Now we have:
\[ \frac{27 \sqrt{490}}{189} \]
We can simplify this fraction:
\[ \frac{27}{189} = \frac{1}{7} \quad \text{(since } 27 \text{ is a common factor)} \]
Thus, we have:
\[ \frac{\sqrt{490}}{7} \]

### Step 5: Further simplify \(\sqrt{490}\)

Now, simplify \(\sqrt{490}\):
\[ 490 = 49 \times 10 = 7^2 \times 10 \implies \sqrt{490} = 7\sqrt{10} \]
Substituting this back gives:
\[ \frac{7\sqrt{10}}{7} = \sqrt{10} \]

### Conclusion

Thus, the final value is:
\[ \sqrt{10} \]