Square Root of 198 - Value, Simplify to 3√22

Quick Reference Table

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.