Square Root of 264 - How to Find the Square Root of 264?

What Is A Square Root?

The square root of a number $n$ is a value $r$ with $r^2 = n$, the number that multiplies by itself to give $n$. If you want the fundamentals first, see what is a square root.

No whole number squares to 264, since $16^2 = 256$ and $17^2 = 289$. So $\sqrt{264}$ lies between 16 and 17, and the exact value is written as the radical $2\sqrt{66}$, a form that comes straight from simplifying radical expressions.

Where Does √264 Appear?

$\sqrt{264}$ turns up whenever an area or a squared distance equals 264, since the side of a square with area 264 square units is $\sqrt{264} = 2\sqrt{66} \approx 16.25$ units. The simplified form $2\sqrt{66}$ is the shape you carry through algebra and geometry, because it keeps the answer exact instead of rounding early. This kind of "pull out the perfect square, leave the rest" step is one of the most common moves in radical arithmetic.

Quick Reference Table

Is The Square Root Of 264 Rational Or Irrational?

$\sqrt{264}$ is irrational. It cannot be written as a fraction of two integers, and its decimal runs on without a repeating block, a property it shares with every root explored under irrational numbers.

The reason is that 264 is not a perfect square. Only perfect squares have rational roots, and although 264 has a perfect-square factor of 4, the leftover 66 has none, so the number as a whole cannot reduce to a rational value. The authoritative definition of the square root operation is given in the Wolfram MathWorld entry on square roots.

How Do You Simplify The Square Root Of 264?

Break 264 into its prime factors, then pull out every matching pair:

$$264 = 2^3 \times 3 \times 11$$

$$264 = (2 \times 2) \times (2 \times 3 \times 11) = 4 \times 66$$

$$\sqrt{264} = \sqrt{4} \times \sqrt{66} = 2\sqrt{66}$$

The single pair of 2s leaves the radical as a factor of 2, and 66 stays inside because $66 = 2 \times 3 \times 11$ has no repeated prime. So $2\sqrt{66}$ is the simplest radical form.

How Do You Find √264?

For the decimal value, long division handles any number that is not a perfect square.

Is 264 A Perfect Square?

No. Its last digit is 4, which keeps it in the running, but the digit sum $2 + 6 + 4 = 12$ reduces to a digital root of 3, and perfect squares never have a digital root of 3. So 264 is confirmed non-square before you divide.

Square Root Of 264 By Long Division

Step 1: The largest square not exceeding 2 is 1, so the first quotient digit is 1, and $2 - 1 = 1$.

Step 2: Bring down 64 to make 164, and double the quotient 1 to get 2.

Step 3: Find a digit $d$ with $(20 + d) \times d \le 164$; here $d = 6$ gives $26 \times 6 = 156$, leaving 8.

Step 4: Add a decimal point, bring down a pair of zeros to make 800, and double 16 to get 32.

Step 5: Find $d$ with $(320 + d) \times d \le 800$; here $d = 2$ gives $322 \times 2 = 644$, leaving 156.

Step 6: Bring down another pair of zeros to make 15600, double 162 to get 324, and $d = 4$ gives $3244 \times 4 = 12976$.

$$\sqrt{264} \approx 16.24$$

Further steps refine the value to $16.248$. Since $2\sqrt{66} = 2 \times 8.124 = 16.248$, the two methods agree, and you can speed up estimates with the nearest-square approach in square root tricks or review the algorithm itself in how to do long division.

Examples Of √264

Example 1

Write $\sqrt{264}$ in simplest radical form.

$$264 = 4 \times 66$$

$$\sqrt{264} = \sqrt{4} \times \sqrt{66} = 2\sqrt{66}$$

Final answer: $2\sqrt{66}$.

Example 2

A student simplifies $\sqrt{264} = 4\sqrt{66}$ after spotting the factor 4. Is that right?

Check by squaring the proposed answer:

$$(4\sqrt{66})^2 = 16 \times 66 = 1056$$

That is 1056, not 264, so the step is wrong. The slip is keeping the 4 instead of taking its square root when it leaves the radical.

Pulling the factor out correctly gives $\sqrt{4} = 2$:

$$\sqrt{264} = 2\sqrt{66}$$

$$(2\sqrt{66})^2 = 4 \times 66 = 264$$

The check now matches. Final answer: $2\sqrt{66}$.

Example 3

Find the decimal value of $\sqrt{264}$ from its simplest form.

$$\sqrt{66} \approx 8.124$$

$$2 \times 8.124 = 16.248$$

Final answer: about $16.248$.

Example 4

Between which two whole numbers does $\sqrt{264}$ lie?

$$16^2 = 256$$

$$17^2 = 289$$

Since $256 < 264 < 289$, the root sits between 16 and 17. Final answer: between 16 and 17.

Example 5

Simplify $\dfrac{\sqrt{264}}{\sqrt{6}}$.

$$\frac{\sqrt{264}}{\sqrt{6}} = \sqrt{\frac{264}{6}} = \sqrt{44}$$

$$\sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11}$$

Final answer: $2\sqrt{11}$.

Common Mistakes

Mistake 1: Keeping the perfect square instead of rooting it

Where it slips in: Pulling the factor 4 out of $\sqrt{264}$.

Don't do this: Writing $\sqrt{264} = 4\sqrt{66}$.

The correct way: The factor leaves the radical as its square root, so $\sqrt{4} = 2$ and the answer is $2\sqrt{66}$. Squaring $2\sqrt{66}$ returns 264, while $4\sqrt{66}$ squares to 1056.

Mistake 2: Assuming √264 cannot be simplified

Where it slips in: Seeing 264 as an awkward three-digit number.

Don't do this: Leaving the answer as a rounded decimal when an exact form is asked for.

The correct way: The rusher who skips the factor check misses that $264 = 4 \times 66$. Always test for a perfect-square factor before deciding a radical will not reduce.

Mistake 3: Splitting the root across addition

Where it slips in: Rewriting 264 as $256 + 8$ to use the nearby perfect square.

Don't do this: Writing $\sqrt{264} = \sqrt{256} + \sqrt{8} = 16 + 2\sqrt{2}$.

The correct way: Roots distribute over multiplication, not addition. Factor 264 into $4 \times 66$ instead, which gives the exact $2\sqrt{66}$.

Conclusion

The square root of 264 is $2\sqrt{66}$, about 16.248, reached by pulling the perfect square 4 out of the radical or by long division. To sharpen radical work with a teacher, explore Bhanzu's algebra tutor, join math classes online, or work with a high school math tutor. Want to practise live? Book a free demo class.

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