The value of $\frac{\sqrt{5043}\times \sqrt{50}}{\sqrt{32}\times \sqrt{1323}}$ is :

The correct option is $\frac{205}{84}$.

To evaluate the given expression step-by-step, we simplify each radical term in the numerator and denominator by prime factoring the numbers under the square root signs.

Given expression:

$\frac{\sqrt{5043}\times \sqrt{50}}{\sqrt{32}\times \sqrt{1323}}$

Step 1: Simplify each square root in the numerator and denominator

• $\sqrt{5043}=\sqrt{1681\times 3}=\sqrt{{41}^2\times 3}=41\sqrt{3}$

• $\sqrt{50}=\sqrt{25\times 2}=\sqrt{5^2\times 2}=5\sqrt{2}$

• $\sqrt{32}=\sqrt{16\times 2}=\sqrt{4^2\times 2}=4\sqrt{2}$

• $\sqrt{1323}=\sqrt{441\times 3}=\sqrt{{21}^2\times 3}=21\sqrt{3}$

Step 2: Substitute the simplified square roots back into the expression

$\frac{41\sqrt{3}\times 5\sqrt{2}}{4\sqrt{2}\times 21\sqrt{3}}$

Step 3: Cancel out common radical factors

Cancel out $\sqrt{3}$ and $\sqrt{2}$ from both the numerator and the denominator:

$\frac{41\times 5}{4\times 21}$

Step 4: Perform the final multiplication

$\frac{205}{84}$

Thus, the value of the given expression is $\frac{205}{84}$.