Square Root of 396 - Value, Simplest Form, and Examples

What Is A Square Root?

The square root of a number $n$ is a value $r$ such that $r^2 = n$, the number that, multiplied by itself, gives $n$. The square root of 396 is the number whose square is 396.

No whole number does this: $19^2 = 361$ (too small) and $20^2 = 400$ (just over). So $\sqrt{396}$ lies between 19 and 20, very close to 20, at about $19.900$.

Because 396 is not a perfect square, $\sqrt{396}$ is an irrational number - but it simplifies neatly, because 396 carries the perfect-square factor 36.

Where √396 Appears In Real Math

$\sqrt{396}$ is the side length of a square whose area is 396 square units - since a square of side $s$ has area $s^2$, solving $s^2 = 396$ gives $s = \sqrt{396} = 6\sqrt{11} \approx 19.9$. It also shows up as a scaled version of $\sqrt{11}$: any shape built to hold 36 times the area of a u nit-$\sqrt{11}$ tile has its edge stretched by a factor of 6, landing on $6\sqrt{11}$.

Is The Square Root Of 396 Rational Or Irrational?

$\sqrt{396}$ is irrational - it cannot be written as a fraction $\frac{p}{q}$ of two integers, and its decimal neither ends nor repeats.

Quick reasoning. A whole number has a rational square root only when it is a perfect square such as 361 or 400. Since 396 sits between $19^2 = 361$ and $20^2 = 400$, it is not a perfect square, so $\sqrt{396}$ is irrational.

Is 396 a perfect square? No. Its prime factorisation is $2^2 \times 3^2 \times 11$, and the prime 11 appears an odd number of times, so no integer squares to 396. Simplifying pulls out the paired factors and leaves the lone 11 behind:

$$\sqrt{396} = \sqrt{36 \times 11} = 6\sqrt{11}$$

The leftover $\sqrt{11}$ is irrational, which is what makes the whole value irrational. Keep $6\sqrt{11}$ in algebra; round to a decimal only when a numeric answer is required.

How Do You Find √396? (Prime Factorisation And Long Division)

Method 1: Prime factorisation (best for the exact form).

Break 396 into primes, then apply the product property of radicals.

$$396 = 4 \times 99$$

$$396 = 2^2 \times 3^2 \times 11$$

Group the paired factors; each pair leaves the radical as a single number.

$$\sqrt{396} = \sqrt{2^2 \times 3^2 \times 11}$$

$$\sqrt{396} = 2 \times 3 \times \sqrt{11}$$

$$\sqrt{396} = 6\sqrt{11}$$

Final answer: $\sqrt{396} = 6\sqrt{11}$.

Method 2: Long division (best for the decimal).

Pair the digits from the decimal point: $3,\overline{96}.\overline{00},\overline{00}$

Step 1: The largest square $\le 3$ is $1^2 = 1$; write 1, remainder $3 - 1 = 2$, bring down 96 to make 296.

Step 2: Double the quotient to get 2; find $d$ with $(20 + d)\times d \le 296$; $d = 9$ gives $29 \times 9 = 261$, remainder 35. Quotient is 19.

Step 3: Bring down 00 to make 3500; double 19 to get 38; $(380 + d)\times d \le 3500$; $d = 8$ gives $388 \times 8 = 3104$, remainder 396. Quotient is 19.8.

Step 4: Bring down 00 to make 39600; double 198 to get 396; $(3960 + d)\times d \le 39600$; $d = 9$ gives $3969 \times 9 = 35721$, remainder 3879. Quotient is 19.89.

Continuing gives $\sqrt{396} \approx 19.900$. The long division never terminates, which is the mark of an irrational value.

Examples Of Square Root Of 396

Example 1

Write $\sqrt{396}$ in simplest radical form.

Find the largest perfect-square factor of 396.

$$396 = 36 \times 11$$

$$\sqrt{396} = \sqrt{36} \times \sqrt{11}$$

$$\sqrt{396} = 6\sqrt{11}$$

The first instinct is to stop at the first perfect square you spot, often 4, and write $2\sqrt{99}$; checking whether 99 still hides a square is the step that gets you all the way to $6\sqrt{11}$.

Final answer: $6\sqrt{11}$.

Example 2

A common slip worth walking through: estimate $\sqrt{396}$ by subtracting roots.

Since $396 = 400 - 4$, the tempting shortcut is to write $\sqrt{396} = \sqrt{400} - \sqrt{4} = 20 - 2 = 18$.

Check it.

$$18^2 = 324$$

But $324 \ne 396$, so 18 is too small - a square root does not split over subtraction any more than over addition.

Bracket instead: $19^2 = 361$ and $20^2 = 400$, so $\sqrt{396}$ sits just below 20, near $19.9$.

Final answer: $\sqrt{396} = 6\sqrt{11} \approx 19.9$, not 18.

Example 3

Evaluate $(\sqrt{396})^2$.

Squaring undoes the square root.

$$(\sqrt{396})^2 = 396$$

Final answer: $396$.

Example 4

Simplify $\sqrt{396} - \sqrt{99}$.

Simplify each radical to reveal like terms.

$$\sqrt{396} = 6\sqrt{11}$$

$$\sqrt{99} = 3\sqrt{11}$$

$$6\sqrt{11} - 3\sqrt{11} = 3\sqrt{11}$$

Radicals combine only when the part under the root matches, exactly like $6x - 3x = 3x$.

Final answer: $3\sqrt{11} \approx 9.95$.

Common Mistakes

Mistake 1: Stopping before the radical is fully simplified

Where it slips in: Factoring out the first perfect square you notice and stopping there.

Don't do this: Writing $\sqrt{396} = 2\sqrt{99}$ and calling it finished.

The correct way: $99$ still holds the perfect square 9, so $2\sqrt{99} = 2 \times 3\sqrt{11} = 6\sqrt{11}$. The habit that fixes this is a second glance at whatever stays under the root, and leaving a hidden perfect square behind is the single most common error on numbers like 396.

Mistake 2: Splitting the root over subtraction

Where it slips in: When the radicand is close to a perfect square, like $396 = 400 - 4$.

Don't do this: Writing $\sqrt{396} = \sqrt{400} - \sqrt{4} = 18$.

The correct way: Square roots do not distribute over addition or subtraction, since $18^2 = 324 \ne 396$. Factor into a product, $396 = 36 \times 11$, to get $6\sqrt{11}$, or bracket between $19^2$ and $20^2$ for a decimal.

Mistake 3: Rounding too early

Where it slips in: Multi-step problems where $\sqrt{396}$ appears mid-calculation.

Don't do this: Replace $\sqrt{396}$ with $19.900$ at the start and carry that through every step.

The correct way: Keep $6\sqrt{11}$ symbolically until the final line, then round once. The exact form is more accurate and makes a subtraction like $6\sqrt{11} - 3\sqrt{11}$ trivial to combine.

Conclusion

The square root of 396 is $6\sqrt{11} \approx 19.900$: irrational, with 36 as its largest perfect-square factor. Pull out the biggest square in one move, double-check that nothing square is left under the root, and keep $6\sqrt{11}$ until the final answer. To practise simplifying radicals with a teacher, explore Bhanzu's algebra tutor, a high school math tutor, or live math classes online. Want the method taught step by step? Book a free demo class.

Read More

• Square Root of 392 — the previous number, simplified to 14√2

• Square Root of 441 — a nearby number that is a perfect square, 21

• Square Root of 245 — the same method with a leftover root 5

• Square Root of 50 — a smaller radical worked to 5√2

• Square Root 1 to 30 — the full reference table of roots in range

• Square Root Tricks — fast estimation methods for roots without a calculator