If $x=\sqrt{198\times 550}$ and $y=\sqrt[3]{99}\times \sqrt[3]{363}$, then $\frac{1}{x}+\frac{1}{y}$ is equal to :

The correct option is 0.03̄.

Given the expressions for $x$ and $y$:

$x=\sqrt{198\times 550}$

$y=\sqrt[3]{99}\times \sqrt[3]{363}$

Step 1: Simplify $x$

Factorize the numbers inside the square root:

$198=2\times 99=2\times 9\times 11=2\times 3^2\times 11$

$550=2\times 275=2\times 25\times 11=2\times 5^2\times 11$

Now, multiply these factorizations:

$198\times 550=(2\times 3^2\times 11)\times (2\times 5^2\times 11)=2^2\times 3^2\times 5^2\times {11}^2$

Taking the square root:

$x=\sqrt{2^2\times 3^2\times 5^2\times {11}^2}=2\times 3\times 5\times 11=330$

Step 2: Simplify $y$

Combine the cube roots:

$y=\sqrt[3]{99\times 363}$

Factorize the numbers:

$99=9\times 11=3^2\times 11$

$363=3\times 121=3\times {11}^2$

Multiply the factorizations:

$99\times 363=(3^2\times 11)\times (3\times {11}^2)=3^3\times {11}^3$

Taking the cube root:

$y=\sqrt[3]{3^3\times {11}^3}=3\times 11=33$

Step 3: Calculate $\frac{1}{x}+\frac{1}{y}$

Substitute the values of $x=330$ and $y=33$:

$\frac{1}{x}+\frac{1}{y}=\frac{1}{330}+\frac{1}{33}$

Find a common denominator:

$\frac{1}{330}+\frac{10}{330}=\frac{11}{330}=\frac{1}{30}$

Convert $\frac{1}{30}$ to a decimal representation:

$\frac{1}{30}=\mathrm{0.0333...}=\mathrm{0.03̄}$

Hence, $\frac{1}{x}+\frac{1}{y}$ is equal to 0.03̄.