Quick Reference Table
Number $n$ | $\sqrt{n}$ (approx.) | Simplest radical | Rational or Irrational |
|---|---|---|---|
196 | 14 | 14 | Rational |
198 | 14.0712 | $3\sqrt{22}$ | Irrational |
200 | 14.1421 | $10\sqrt{2}$ | Irrational |
208 | 14.4222 | $4\sqrt{13}$ | Irrational |
220 | 14.8324 | $2\sqrt{55}$ | Irrational |
225 | 15 | 15 | Rational |
Where Does √198 Appear?
$\sqrt{198}$ is the side length of a square with an area of 198 square units, a value just past 14. It also surfaces in the Pythagorean theorem whenever the sum of two squared legs lands on 198 - for example, a right triangle with legs $3\sqrt{2}$ and $12$, since $18 + 180 = 198$.
What Is a Square Root?
The square root of a number $n$ is the value $r$ for which $r^2 = n$. The square root of 198 is the number that, multiplied by itself, gives 198.
No integer works. Since $14^2 = 196$ is just under and $15^2 = 225$ is well over, $\sqrt{198}$ lies between 14 and 15, barely above 14.
The symbol $\sqrt{198}$ means the principal (positive) root, even though $(-14.0712)^2$ also equals 198.
Is the Square Root of 198 Rational or Irrational?
$\sqrt{198}$ is irrational - it cannot be written as a fraction of two integers, and its decimal never terminates or repeats.
Why? A whole number has a rational square root only when it is a perfect square. 198 is not a perfect square, so $\sqrt{198}$ is irrational even after it simplifies to $3\sqrt{22}$.
Can the square root of 198 be simplified? Yes. Unlike a prime-heavy number, 198 carries the perfect-square factor 9, so it reduces to $3\sqrt{22}$ - the kind of reduction practised across simplifying radical expressions.
How Do You Find √198? (Prime Factorization and Long Division)
Prime factorization does the simplifying.
$$198 = 2 \times 3 \times 3 \times 11$$
$$198 = 2 \times 3^2 \times 11$$
The pair of 3s forms a perfect square, so a 3 leaves the radical.
$$\sqrt{198} = \sqrt{3^2 \times 22} = 3\sqrt{22}$$
Because $22 = 2 \times 11$ has no repeated prime, nothing else comes out. The square root 1 to 30 hub lists which nearby numbers reduce this way and which do not.
Long division gives the decimal. Pair the digits: $1,98.\overline{00},\overline{00}$.
Step 1: Find the largest integer whose square is $\leq 198$.
$$14^2 = 196 \leq 198$$
Step 2: Subtract and bring down the first pair of zeros.
$$198 - 196 = 2$$
$$\text{new dividend} = 200$$
Step 3: Double the quotient 14 to get 28, then find a digit $d$ with $(280 + d) \times d \leq 200$.
$$280 \times 0 = 0$$
So $d = 0$, and the quotient reads 14.0.
Step 4: Bring down the next pair.
$$\text{new dividend} = 20000$$
Step 5: Double 140 to get 280, then find $d$ with $(2800 + d) \times d \leq 20000$.
$$2807 \times 7 = 19649$$
Step 6: Continuing one more round gives:
$$\sqrt{198} \approx 14.0712$$
For a quick check without long division, the estimate in square root tricks places $\sqrt{198}$ just above $\sqrt{196} = 14$.
Examples of √198
Example 1
Confirm that $\sqrt{198} = 3\sqrt{22}$ numerically.
$$3\sqrt{22} = 3 \times 4.6904 = 14.0712$$
This matches the long-division value.
Final answer: $\sqrt{198} = 3\sqrt{22} \approx 14.0712$.
Example 2
Simplify $\sqrt{198}$ from the factorization $198 = 9 \times 22$.
The first instinct is to move the whole 9 outside the radical:
$$\sqrt{198} = 9\sqrt{22} \quad ?$$
Check it. $9\sqrt{22} \approx 9 \times 4.69 = 42.2$, but $\sqrt{198} \approx 14.07$. Far too big.
The slip is pulling out the factor 9 instead of its square root. What leaves a radical is $\sqrt{9} = 3$, not 9 itself.
Final answer: $\sqrt{198} = 3\sqrt{22}$.
Example 3
Between which two consecutive integers does $\sqrt{198}$ lie?
$$14^2 = 196$$
$$15^2 = 225$$
Since $196 < 198 < 225$, the root is between 14 and 15, and because 198 is very close to 196 it sits just above 14.
Final answer: between 14 and 15, near 14.07.
Example 4
Simplify $\sqrt{2} \times \sqrt{99}$.
Combine under one radical.
$$\sqrt{2} \times \sqrt{99} = \sqrt{198}$$
$$\sqrt{198} = 3\sqrt{22}$$
Final answer: $3\sqrt{22} \approx 14.0712$.
Example 5
A square courtyard covers 198 square metres. What is its side, in simplest radical form?
Side length $= \sqrt{\text{area}}$.
$$\text{side} = \sqrt{198} = 3\sqrt{22} \approx 14.07 \text{ m}$$
Final answer: $3\sqrt{22}$ m, about $14.07$ m.
Common Mistakes
Mistake 1: Stopping the simplification too soon
Where it slips in: When a student factors $198 = 2 \times 99$ and writes $\sqrt{2} \times \sqrt{99}$, then stops.
Don't do this: Leaving the answer as $\sqrt{2}\sqrt{99}$, unaware that 99 still contains the perfect square 9.
The correct way: Factor all the way down to primes. Then $99 = 9 \times 11$ reveals the perfect square, giving $3\sqrt{22}$. The habit of factoring only halfway is what a full prime-factor tree fixes.
Mistake 2: Extracting the factor instead of its root
Where it slips in: When simplifying $\sqrt{9 \times 22}$.
Don't do this: Writing $9\sqrt{22}$.
The correct way: A perfect square leaves the radical as its root: $\sqrt{9} = 3$, so the answer is $3\sqrt{22}$.
Mistake 3: Treating 198 as a perfect square
Where it slips in: When a quick glance near 200 suggests a whole-number root.
Don't do this: Writing $\sqrt{198} = 14$.
The correct way: $14^2 = 196$, not 198, so $\sqrt{198}$ is irrational and equals $3\sqrt{22}$.
Conclusion
The square root of 198 simplifies to $3\sqrt{22}$, roughly $14.0712$, and stays irrational because 198 is not a perfect square. To practise radical simplification with a teacher, explore Bhanzu's algebra tutor or a high school math tutor, or join structured math classes online. Want to see the factor-tree method worked live? Book a free demo class.
Read More
Square Root of 12 — a smaller root that simplifies to $2\sqrt{3}$ the same way.
Square Root of 50 — reduces to $5\sqrt{2}$, another perfect-square extraction.
Square Root of 196 — the perfect square just below 198.
Squares and Square Roots — the full relationship between a number and its root.
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