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

Question

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

Accepted Answer

To determine why Option 2 is the correct answer, we need to calculate the values of $ x $ and $ y $ and then find $\frac{1}{x} + \frac{1}{y}$.

1. **Calculate $ x $:**

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

We can simplify this expression by multiplying under the square root:

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

Calculating the product:

$$
   198 \times 550 = 108900
   $$

Therefore:

$$
   x = \sqrt{108900} = 330
   $$

2. **Calculate $ y $:**

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

Similarly, we can simplify this expression by multiplying under the cube root:

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

Calculating the product:

$$
   99 \times 363 = 35937
   $$

Therefore:

$$
   y = \sqrt[3]{35937} = 33
   $$

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

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

To add these fractions, find a common denominator. The least common multiple of 330 and 33 is 330.

$$
   \frac{1}{330} + \frac{1}{33} = \frac{1}{330} + \frac{10}{330} = \frac{11}{330}
   $$

Simplifying $\frac{11}{330}$:

$$
   \frac{11}{330} = \frac{1}{30} = 0.0\overline{3}
   $$

Thus, the value of $\frac{1}{x} + \frac{1}{y}$ is $0.0\overline{3}$, which corresponds to Option 2.

**Explanation for Incorrect Options:**

- **Option 1 (0.$\overline{01}$)**: This option suggests a repeating decimal of 0.01, which is significantly smaller than the calculated value of 0.0$\overline{3}$.

- **Option 3 (0.$\overline{33}$)**: This option suggests a repeating decimal of 0.33, which is much larger than the calculated value of 0.0$\overline{3}$.

- **Option 4 (0.$\overline{1}$)**: This option suggests a repeating decimal of 0.1, which is also larger than the calculated value of 0.0$\overline{3}$.

Therefore, based on the calculations and logical reasoning, Option 2 is indeed the correct answer.