Square Root of 729 - Value, Methods, and Examples

What Is A Square Root?

The square root of a number $n$ is a value $r$ with $r^2 = n$. So $\sqrt{729}$ is the number that, multiplied by itself, gives $729$. For the underlying idea, see what a square root is.

Here an integer works exactly: $27 \times 27 = 729$. That makes $729$ a member of the perfect squares, and its square root a whole number rather than a decimal.

Where √729 Appears

The square root of 729 is the side length of a square whose area is $729$ square units: a $27 \times 27$ grid holds exactly $729$ cells. The number itself is special because $729 = 3^6$, so it also appears as a perfect cube ($9^3 = 729$) and as the count of the smallest cells in a $9 \times 9 \times 9$ cube. Numbers that are both a perfect square and a perfect cube, like $729$ and $64$, are sixth powers, and they are the reason $\sqrt{729}$ comes out clean.

Quick Reference Table

Is The Square Root Of 729 Rational Or Irrational?

$\sqrt{729}$ is rational. It equals the integer $27$, which is the ratio $\frac{27}{1}$, so it is rational by definition.

A whole number has a rational square root precisely when it is a perfect square. The prime factorization $729 = 3^6$ has an even exponent, which means the primes pair up perfectly: $3^6 = (3^3)^2 = 27^2$. Because every prime factor forms a complete pair, nothing is left stuck under the radical, and the answer is the exact whole number $27$. This is the same principle behind squares and square roots generally, and Britannica's overview of the square root states the rule the same way.

How Do You Find The Square Root Of 729?

Two hand methods both land on $27$.

Method 1: Prime factorization

Step 1: Break $729$ into primes.

$729 = 3 \times 243$

$= 3 \times 3 \times 81$

$= 3^6$

Step 2: Pair the identical primes.

$3^6 = (3 \times 3 \times 3) \times (3 \times 3 \times 3)$

Step 3: Take one factor from each pair out of the radical.

$\sqrt{729} = 3 \times 3 \times 3 = 27$

Final answer: $\sqrt{729} = 27$.

Method 2: Long division

Step 1: Group the digits in pairs from the right: $\overline{7},\overline{29}$.

Step 2: For the first group $7$, the greatest square not exceeding it is $2^2 = 4$. Write $2$ as the first quotient digit.

Step 3: Subtract: $7 - 4 = 3$. Bring down $29$ to make $329$.

Step 4: Double the quotient: $2 \times 2 = 4$. Find a digit $d$ with $(40 + d) \times d \leq 329$. Test $d = 7$: $47 \times 7 = 329$ exactly.

Step 5: The remainder is $0$, so the process ends. The quotient is $27$.

$\sqrt{729} = 27$

Examples Of The Square Root Of 729

Example 1

Verify that 27 is the square root of 729.

$27 \times 27 = 729$

So $\sqrt{729} = 27$ is confirmed.

Example 2 (Wrong path first)

Find $\sqrt{729}$ quickly.

Wrong attempt. A student halves the number, writing $\sqrt{729} = 729 \div 2 = 364.5$, mixing up "square root" with "divide by two".

The break. Check it: $364.5^2 = 132{,}860.25$, nowhere near $729$. Dividing by two is not the same operation as taking a square root.

Correct. A square root asks which number times itself gives $729$. Since $27 \times 27 = 729$, the answer is $\sqrt{729} = 27$.

Example 3

Solve $x^2 = 729$.

$x^2 = 729$

$x = \pm\sqrt{729}$

$x = \pm 27$

Both $27$ and $-27$ satisfy the equation.

Example 4

Simplify $\sqrt{\dfrac{729}{4}}$.

Split the radical over the fraction.

$\sqrt{\dfrac{729}{4}} = \dfrac{\sqrt{729}}{\sqrt{4}}$

$= \dfrac{27}{2}$

$= 13.5$

Example 5

A square courtyard has an area of 729 square metres. Find its perimeter.

Side $= \sqrt{729} = 27$ m

Perimeter $= 4 \times 27 = 108$ m

The courtyard is $27$ m on each side, so its perimeter is $108$ metres.

Common Mistakes

Mistake 1: Assuming 729 is not a perfect square

Where it slips in: Reaching for a decimal because $729$ looks awkward.

Don't do this: Reporting $\sqrt{729} \approx 27.0$ as an approximation, as if the root were irrational.

The correct way: Check the factorization first. $729 = 3^6 = 27^2$, so the root is the exact integer $27$. Students who scan for a perfect-square factorization before estimating never over-round a clean root.

Mistake 2: Confusing the square root with the cube root

Where it slips in: $729$ is both a perfect square and a perfect cube.

Don't do this: Writing $\sqrt{729} = 9$, which is actually the cube root.

The correct way: $\sqrt{729} = 27$ (since $27^2 = 729$), while the cube root is $\sqrt[3]{729} = 9$ (since $9^3 = 729$). Read the index carefully before answering.

Mistake 3: Reporting only one root when solving an equation

Where it slips in: Solving $x^2 = 729$.

Don't do this: Writing $x = 27$ and stopping.

The correct way: The equation $x^2 = 729$ has two solutions, $x = \pm 27$. The symbol $\sqrt{729}$ alone means only the positive root, $27$.

Conclusion

The square root of 729 is exactly $27$, a rational whole number, because $729 = 3^6 = 27^2$ is a perfect square, and its even prime exponents pair up with nothing left over. It is also a perfect cube, which is why both its square root and cube root come out clean. To strengthen this with a teacher, explore Bhanzu's algebra tutor sessions, get help with algebra, or join live math classes online.

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Read More

• Square Root of 196 — another perfect square, $14^2$.

• Square Root of 441 — the perfect square $21^2$, close in size.

• Perfect Cube — why 729 is also a cube, and how cubes are built.

• Simplifying Radical Expressions — the method for reducing any root.

• Square Root Tricks — fast ways to spot and estimate square roots.

• Square Root 1 to 25 — the reference table of common square roots.