Simplify the square root of 198.

Step-by-step solution

1. 1

   Factor the radicand into primes. $198$ is even, so start there:

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

2. 2

   Identify the perfect-square part. Only the $3^2$ appears as a repeated prime, so the largest perfect-square factor is $9$:

   $198=9\times 22$

3. 3

   Split and simplify.

   $\sqrt{198}=\sqrt{9}\cdot\sqrt{22}=3\sqrt{22}$

4. 4

   Verify that $22$ cannot be reduced further. $22=2\times 11$ with both primes appearing once, so it is square-free and $3\sqrt{22}$ is the final simplified form. A common wrong move is to write $\sqrt{198}=\sqrt{2}\sqrt{99}$ and stop — that hides the square factor inside $99$.

5. 5

   Check numerically. $\sqrt{22}\approx 4.6904158$, so $3\sqrt{22}\approx 14.0712474$, and $14.0712474^2=198.000$ $\checkmark$.