Square Root Of 324: Value, Methods And Worked Examples

What Is A Square Root?

A square root of a number $n$ is a number $r$ with $r \times r = n$. For 324, both $18 \times 18$ and $(-18) \times (-18)$ give 324.

The radical sign $\sqrt{\phantom{x}}$ always means the positive one, called the principal square root. So $\sqrt{324} = 18$, while the two square roots of 324 together are written $\pm 18$.

Where Does The Square Root Of 324 Show Up, And Why Does It Matter?

The square root of 324 shows up whenever a square has an area of 324 square units, since its side is then 18. A square tile 18 inches on each side covers 324 square inches, which is exactly 2.25 square feet.

A standard Go board is a $19 \times 19$ grid of lines with 361 points, as Wikipedia's Go article notes. Those lines enclose an $18 \times 18$ block of small squares, so the board holds 324 squares, and $\sqrt{324} = 18$ squares run along each side.

Quick Reference Table

These are the square roots just below and above 324, with simplified forms and four-decimal values.

The perfect squares 289, 324 and 361 sit on consecutive whole numbers, 17, 18 and 19. Every number strictly between them has an irrational root, which is why $\sqrt{323}$ and $\sqrt{325}$ need decimals while $\sqrt{324}$ does not.

Is 324 A Perfect Square?

Yes. 324 is a perfect square because it equals $18^2$, the square of a whole number.

You can justify it without multiplying anything out. In the prime factorization $324 = 2^2 \times 3^4$, every exponent is even, and that is exactly the condition for a whole number to be one of the perfect squares.

324 is not a perfect cube, though. It sits between $6^3 = 216$ and $7^3 = 343$, and its cube root is about 6.8683.

Is The Square Root Of 324 Rational Or Irrational?

The square root of 324 is rational. It equals 18, which can be written as the fraction $\frac{18}{1}$, and that is the definition of a rational number.

This is the general rule at work: the square root of a whole number is rational only when the number is a perfect square. Change 324 by one in either direction and the root turns irrational, as the table shows for 323 and 325.

How Do You Find The Square Root Of 324?

Three hand methods give 18, and a fourth method checks it. Each one shows a different reason the answer is a whole number.

Method 1: Prime Factorization

Split 324 into prime factors, pair equal primes, and take one prime from each pair.

$$324 = 2 \times 162$$

$$= 2 \times 2 \times 81$$

$$= 2 \times 2 \times 3 \times 3 \times 3 \times 3$$

$$= (2 \times 2) \times (3 \times 3) \times (3 \times 3)$$

$$\sqrt{324} = 2 \times 3 \times 3 = 18$$

In exponent form, $324^{1/2} = (2^2 \times 3^4)^{1/2} = 2 \times 3^2 = 18$, the same move written with exponents.

Method 2: Long Division

The long division method pairs digits from the right, so 324 becomes $3 \mid 24$.

The remainder is 0 with no pairs left, so $\sqrt{324} = 18$ exactly. Bringing down a pair of zeros would only add zeros after the decimal point.

Method 3: Repeated Subtraction

Subtract the odd numbers 1, 3, 5, 7, and so on from 324 until you reach 0, then count the steps. This works because the sum of odd numbers from 1 up to the $n$th odd number is $n^2$.

• $324 - 1 = 323$, $323 - 3 = 320$, $320 - 5 = 315$, $315 - 7 = 308$

• $308 - 9 = 299$, $299 - 11 = 288$, $288 - 13 = 275$, $275 - 15 = 260$

• $260 - 17 = 243$, $243 - 19 = 224$, $224 - 21 = 203$, $203 - 23 = 180$

• $180 - 25 = 155$, $155 - 27 = 128$, $128 - 29 = 99$, $99 - 31 = 68$

• $68 - 33 = 35$, $35 - 35 = 0$

That is 18 subtractions, so $\sqrt{324} = 18$. The method is slow for large numbers but makes the meaning of a perfect square visible.

Method 4: Estimate, Then Check With Heron's Method

324 ends in 4, so its root must end in 2 or 8, because only $2^2$ and $8^2$ end in 4. Since $17^2 = 289$ and $19^2 = 361$, the root lies between 17 and 19, leaving 18 as the only candidate.

Heron's method, described by Hero of Alexandria around AD 60, confirms it from any rough guess. Start at 20:

$$x_1 = \frac{1}{2}\left(20 + \frac{324}{20}\right) = 18.1$$

$$x_2 = \frac{1}{2}\left(18.1 + \frac{324}{18.1}\right) = 18.0003$$

Two steps land within 0.0003 of 18, and a calculator's square root key returns exactly 18.

Can You Simplify The Square Root Of 324?

Yes, all the way to a whole number. Simplifying a radical means pulling perfect-square factors out of the root, and in 324 every factor is part of a pair.

$$\sqrt{324} = \sqrt{4 \times 81} = \sqrt{4} \times \sqrt{81} = 2 \times 9 = 18$$

Compare $\sqrt{162}$, half of 324, which simplifies only to $9\sqrt{2}$ because one factor of 2 has no partner. The simplifying radical expressions guide covers the general method.

What Is The Positive Square Root Of 324?

The positive square root of 324 is 18, and the negative square root is $-18$. The equation $x^2 = 324$ has both as solutions:

$$x^2 = 324$$

$$x = \pm\sqrt{324}$$

$$x = 18 \quad \text{or} \quad x = -18$$

The symbol $\sqrt{324}$ on its own never means $-18$. The $\pm$ appears only when you solve an equation in which $x$ is squared.

Examples Of Square Root Of 324

Example 1

Evaluate $\sqrt{324} + \sqrt{81}$.

$$\sqrt{324} + \sqrt{81} = 18 + 9 = 27$$

Example 2

A square garden has an area of 324 square metres. How long is each side?

Wrong path: divide the area by 4, as if it were the perimeter.

$$324 \div 4 = 81$$

Check: a square with side 81 m has area $81^2 = 6561$ square metres, about twenty times too big. Dividing by 4 finds a side from a perimeter, not from an area.

Correct path: the side of a square is the square root of its area.

$$s = \sqrt{324} = 18\ \text{m}$$

Example 3

Solve $2x^2 = 648$.

$$x^2 = \frac{648}{2} = 324$$

$$x = \pm\sqrt{324}$$

$$x = \pm 18$$

Example 4

A right triangle has legs 18 cm and 24 cm. Find the hypotenuse.

By the Pythagoras theorem:

$$c^2 = 18^2 + 24^2 = 324 + 576 = 900$$

$$c = \sqrt{900} = 30\ \text{cm}$$

Here 324 appears as $18^2$; recognising it on sight saves a multiplication.

Example 5

Find $\sqrt{32400}$ and $\sqrt{0.0324}$.

$$\sqrt{32400} = \sqrt{324} \times \sqrt{100} = 18 \times 10 = 180$$

$$\sqrt{0.0324} = \sqrt{\frac{324}{10000}} = \frac{18}{100} = 0.18$$

Moving the decimal point two places in 324 moves it one place in the root. The first instinct is to move it two places in the answer as well, giving 1.8 for $\sqrt{0.0324}$, and squaring 1.8 to get 3.24 exposes that straight away.

Example 6

A floor 9 ft by 12 ft is covered with square tiles of area 324 square inches. How many tiles are needed?

$$\text{tile side} = \sqrt{324} = 18\ \text{in}$$

$$9\ \text{ft} = 108\ \text{in}, \quad 108 \div 18 = 6$$

$$12\ \text{ft} = 144\ \text{in}, \quad 144 \div 18 = 8$$

$$6 \times 8 = 48\ \text{tiles}$$

Where Do Students Go Wrong With The Square Root Of 324?

Mistake 1: Halving Instead Of Rooting

Where it slips in: Mental work under time pressure.

Don't do this: Writing $\sqrt{324} = 162$.

The correct way: A square root asks which number times itself gives 324, and $162 \times 162$ is far larger. The rusher hears "root" and reaches for "divide by 2"; the habit that stops it is squaring the answer before writing it down.

Mistake 2: Writing $\sqrt{324} = \pm 18$

Where it slips in: Right after solving equations like $x^2 = 324$.

Don't do this: Attaching $\pm$ to the radical symbol itself.

The correct way: $\sqrt{324} = 18$ only. The two ideas sit side by side in the same lesson, the number $\sqrt{324}$ and the equation $x^2 = 324$, and students who memorised "square roots have two answers" apply it to the symbol, where it does not belong.

Mistake 3: Pairing Digits From The Left

Where it slips in: The first step of the long division method.

Don't do this: Splitting 324 as $32 \mid 4$, which starts the root at 5 because $5^2 = 25$.

The correct way: Pair from the decimal point leftwards: $3 \mid 24$. The first group may have one digit, and the root of a three-digit number always has two digits.

Where Can You Get Help With Square Roots?

If perfect squares like 324 still take a long time to spot, practice with a teacher makes the patterns stick.

An algebra tutor or a middle school math tutor can work through factor trees and long division with you, and mental maths for kids builds the quick recognition of squares from $11^2$ to $25^2$. Structured math classes online cover roots, exponents and the algebra that comes next.

Conclusion

• The square root of 324 is exactly 18, because $18 \times 18 = 324$.

• 324 is a perfect square since $324 = 2^2 \times 3^4$ has only even exponents, so its root is rational.

• Prime factorization, long division and repeated subtraction all give 18, and Heron's method confirms it from any guess.

• $\sqrt{324}$ means only 18, while $x^2 = 324$ has two solutions, $18$ and $-18$.

• The most common errors are halving 324 and pairing digits from the left in long division.

Practice These Before Moving On

1. Find $\sqrt{324} \times \sqrt{4}$.

2. Solve $x^2 - 324 = 0$.

3. Find $\sqrt{3.24}$.

4. A square photo has area 324 square centimetres. Find its perimeter.

5. Use prime factorization to find $\sqrt{1296}$, given $1296 = 324 \times 4$.

Answers:

1. $18 \times 2 = 36$

2. $x = 18$ or $x = -18$

3. $1.8$

4. Side $18$ cm, perimeter $4 \times 18 = 72$ cm

5. $1296 = 2^4 \times 3^4$, so $\sqrt{1296} = 2^2 \times 3^2 = 36$

If question 5 stalls, go back to Method 1 and circle the prime pairs again. Practice these to solidify your understanding, and if you would like a teacher to guide you, Book a free demo class.

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