Square Root of 108: Simplified 6√3 and Value 10.3923

What Is A Square Root?

A square root of a number $n$ is a number $r$ with $r^2 = n$. Every positive number has two square roots, one positive and one negative, and the symbol $\sqrt{n}$ always means the positive one (the principal square root).

So $\sqrt{108}$ asks for the positive number that, multiplied by itself, gives 108. No whole number does this, which is the first clue that the answer will be irrational.

Where Does The Square Root Of 108 Appear, And Why Does It Matter?

$\sqrt{108}$ is the height of an equilateral triangle with side 12. Drop the altitude, and the Pythagoras theorem gives $\sqrt{12^2 - 6^2} = \sqrt{144 - 36} = \sqrt{108} = 6\sqrt{3}$.

The same number is the space diagonal of a cube with edge 6, since $\sqrt{6^2 + 6^2 + 6^2} = \sqrt{108}$. That is why $\sqrt{108}$ keeps turning up in packing, tiling and box-design problems, and why recognising it as $6\sqrt{3}$ on sight saves real time.

Quick Reference Table

The neighbours of 108 show where its root sits. Only 100 and 121 are perfect squares in this stretch, so every other entry is irrational.

Notice how slowly the roots climb. Moving from 100 to 121 adds 21 to $n$ but only 1 to the root, so each step of 1 in $n$ moves the root by roughly $0.05$.

Is The Square Root Of 108 Rational Or Irrational?

The square root of 108 is irrational: it cannot be written as a fraction $\frac{p}{q}$ of two integers, and its decimal never terminates or repeats.

A whole number has a rational square root only when it is one of the perfect squares. 108 is not, and there is a ten-second test for that: a perfect square can only end in 0, 1, 4, 5, 6 or 9, and 108 ends in 8.

Proof sketch: Since $\sqrt{108} = 6\sqrt{3}$, suppose $6\sqrt{3} = \frac{p}{q}$ for integers $p$ and $q$. Then $\sqrt{3} = \frac{p}{6q}$, a fraction, which contradicts the known fact that root 3 is irrational.

A rational number times an irrational one (other than zero) is always irrational, and that single line is the whole argument. It is also why $\sqrt{108}$ belongs to the irrational numbers even though $36$, the big factor inside it, is a perfect square.

How Do You Simplify The Square Root Of 108?

Yes, the square root of 108 can be simplified: pull out the largest perfect-square factor, which is 36. The result is $6\sqrt{3}$, and that is its simplest radical form.

$$\sqrt{108} = \sqrt{36 \times 3}$$

$$= \sqrt{36} \times \sqrt{3}$$

$$= 6\sqrt{3}$$

The perfect-square factors of 108 are 1, 4, 9 and 36. Picking a smaller one still works, but it leaves more to do: $\sqrt{108} = 2\sqrt{27} = 2 \times 3\sqrt{3} = 6\sqrt{3}$.

A radical is fully simplified when the number under the root has no perfect-square factor left other than 1. Here 3 is prime, so $6\sqrt{3}$ is finished (the general routine is covered in simplifying radical expressions).

$\sqrt{108}$ is one member of a tidy family: every number of the form $3k^2$ has square root $k\sqrt{3}$.

Once you know the square root of 3 is about $1.7321$, the whole column follows by multiplication: $6 \times 1.7321 = 10.3926$, which is within $0.0003$ of the true value.

How Do You Find The Square Root Of 108?

Four methods give the square root of 108, and each answers a different need: prime factorization gives the exact form, long division gives guaranteed decimal digits, estimation gives a fast mental answer, and the Babylonian method converges fastest.

Method 1: Prime Factorization

Break 108 into prime numbers, then pair equal factors.

$$108 = 2 \times 54 = 2 \times 2 \times 27 = 2 \times 2 \times 3 \times 3 \times 3$$

$$108 = 2^2 \times 3^3 = 2^2 \times 3^2 \times 3$$

Each pair leaves the root as a single factor, so $\sqrt{108} = 2 \times 3 \times \sqrt{3} = 6\sqrt{3}$. The unpaired 3 stays inside.

Method 2: Long Division

The long division method produces one correct digit at a time. Pair the digits from the decimal point outward: 108 becomes 1 | 08, and each decimal place adds a pair 00.

Each new trial divisor is twice the root so far (ignoring the decimal point) with the next digit appended. For example, at step 4 the root so far is 103, twice that is 206, and $2069 \times 9 = 18621$ is the largest product that fits under 19100.

The next digit comes out as 0 and the one after as 4, so $\sqrt{108} = 10.39230\ldots$, which rounds to $10.3923$.

Method 3: Estimation Between 100 And 121

108 sits between 100 and 121, so $\sqrt{108}$ sits between 10 and 11. Linear interpolation gives a first estimate.

$$\sqrt{108} \approx 10 + \frac{108 - 100}{121 - 100} = 10 + \frac{8}{21} \approx 10.381$$

The true value is $10.3923$, so this shortcut is off by about $0.011$. It always lands slightly low, because the square-root curve bends downward between perfect squares.

Method 4: Babylonian (Heron's) Method

Start with a guess $x_0$, then average the guess with 108 divided by the guess. This is the method Heron of Alexandria described in his Metrica, and it is the same calculation as Newton's method applied to $x^2 - 108 = 0$.

$$x_{n+1} = \frac{1}{2}\left(x_n + \frac{108}{x_n}\right)$$

• Start: $x_0 = 10$

• Step 1: $x_1 = \frac{1}{2}(10 + 10.8) = 10.4$

• Step 2: $x_2 = \frac{1}{2}\left(10.4 + \frac{108}{10.4}\right) = 10.392308\ldots$

• Step 3: $x_3 = 10.3923048454\ldots$

Two steps already give four correct decimal places, and the third is correct to ten. The number of correct digits roughly doubles each round.

Method 5: Calculator Check

A calculator returns $\sqrt{108} = 10.392304845\ldots$, which matches every method above. Check the rounded value by squaring it: $10.3923^2 = 107.9999$, just under 108, as it should be.

Is The Square Root Of 108 Positive Or Negative?

The symbol $\sqrt{108}$ means the positive value, $6\sqrt{3} \approx 10.3923$. Both $6\sqrt{3}$ and $-6\sqrt{3}$ square to 108, so the equation $x^2 = 108$ has two solutions, written $x = \pm 6\sqrt{3}$.

The $\pm$ belongs to solving an equation, not to the radical sign itself. Writing $\sqrt{108} = \pm 10.39$ mixes the two ideas.

Examples Of Square Root Of 108

Example 1

Simplify $3\sqrt{108}$.

$$3\sqrt{108} = 3 \times 6\sqrt{3}$$

$$= 18\sqrt{3}$$

$$\approx 31.1769$$

Example 2

Find $\sqrt{108}$ by writing $108 = 100 + 8$.

The tempting move is to split the root across the sum.

$$\sqrt{100 + 8} \overset{\text{wrong}}{=} \sqrt{100} + \sqrt{8} = 10 + 2\sqrt{2} \approx 12.83$$

Square it as a test: $12.83^2 \approx 164.6$, nowhere near 108. A square root splits across a product, never across a sum, so rewrite 108 as a product instead.

$$\sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3} \approx 10.3923$$

Example 3

Simplify $\sqrt{108} + \sqrt{12} - \sqrt{27}$.

Simplify each radical first so the terms share $\sqrt{3}$.

$$6\sqrt{3} + 2\sqrt{3} - 3\sqrt{3}$$

$$= (6 + 2 - 3)\sqrt{3}$$

$$= 5\sqrt{3} \approx 8.6603$$

Example 4

Evaluate $\sqrt{108} \div \sqrt{75}$.

$$\frac{\sqrt{108}}{\sqrt{75}} = \frac{6\sqrt{3}}{5\sqrt{3}}$$

$$= \frac{6}{5}$$

$$= 1.2$$

The irrational parts cancel, leaving a rational answer.

Example 5

Solve $2x^2 = 216$.

$$x^2 = 108$$

$$x = \pm\sqrt{108}$$

$$x = \pm 6\sqrt{3} \approx \pm 10.3923$$

This is the simplest kind of quadratic equation, and both signs count as solutions.

Example 6

A square floor tile has area $108\ \text{cm}^2$. Find its side and its diagonal.

$$\text{side} = \sqrt{108} = 6\sqrt{3} \approx 10.39\ \text{cm}$$

$$\text{diagonal} = \text{side} \times \sqrt{2} = 6\sqrt{3} \times \sqrt{2} = 6\sqrt{6}$$

$$6\sqrt{6} \approx 14.70\ \text{cm}$$

Keeping the exact form until the last line avoids rounding error: $10.39 \times 1.414$ gives $14.69$, already off in the second decimal place.

Where Do Students Go Wrong With $\sqrt{108}$?

Mistake 1: Stopping At A Partial Simplification

Where it slips in: Simplifying by spotting the first perfect square that comes to mind, usually 4 or 9.

Don't do this: Writing $\sqrt{108} = 2\sqrt{27}$ or $3\sqrt{12}$ and calling it simplified.

The correct way: Check the number left under the root. Both 27 and 12 still hide a perfect square, so keep going until you reach $6\sqrt{3}$, or start with the largest factor, 36.

The student who finds 4 first and writes $2\sqrt{27}$ usually stops because the answer looks finished. Asking "does the number under the root still have a square factor?" before moving on fixes it every time.

Mistake 2: Pairing Digits From The Wrong End

Where it slips in: Setting up long division for an odd-digit number like 108.

Don't do this: Pairing from the left as 10 | 8, which makes the first digit 3 and quietly computes $\sqrt{10.8} \approx 3.286$ instead.

The correct way: Pair from the decimal point outward: 1 | 08. A quick estimate first (between 10 and 11) catches the error at once, because a two-digit whole part was expected.

The rusher who skips the estimate is the one who hands in $3.29$ without a second look. Bracketing between 100 and 121 takes five seconds and protects the whole calculation.

Mistake 3: Writing $\pm$ In Front Of The Radical Value

Where it slips in: Moving between "solve $x^2 = 108$" and "find $\sqrt{108}$" in the same exercise set.

Don't do this: Writing $\sqrt{108} = \pm 10.39$.

The correct way: $\sqrt{108}$ is the principal root, $10.3923$. The $\pm$ appears only when solving $x^2 = 108$.

This one confused me too when I first taught it, because many older textbooks print "the square roots of 108" and "$\sqrt{108}$" a line apart. They are different requests, and the radical sign settles which one is meant.

Where Can You Get Help With Square Roots And Radicals?

Square roots feel mechanical until simplifying radicals, the $\pm$ rule and long division all have to work together in one problem. If that is where marks are being lost, an algebra tutor can rebuild the reasoning behind each step, and help with algebra sessions cover radicals alongside exponents.

For learners meeting square roots for the first time, a middle school math tutor fits the level, and math classes online keep the practice regular.

Conclusion

• The square root of 108 is $6\sqrt{3}$, about $10.3923$, and it is irrational.

• Prime factorization ($108 = 2^2 \times 3^3$) gives the exact form; pull out the largest perfect-square factor, 36.

• Long division, estimation between 100 and 121, and the Babylonian method all confirm the decimal $10.3923$.

• $\sqrt{108}$ is the height of an equilateral triangle with side 12 and the space diagonal of a cube with edge 6.

• The most common errors are stopping at $2\sqrt{27}$, splitting the root across $100 + 8$, and attaching $\pm$ to the radical value.

Practice These Before Moving On

1. Simplify $2\sqrt{108}$.

2. Simplify $\sqrt{108} \times \sqrt{3}$.

3. Find $\sqrt{108}$ to the nearest tenth without a calculator.

4. Rationalise $\frac{12}{\sqrt{108}}$.

5. Simplify $\frac{\sqrt{108}}{\sqrt{27}}$.

Answers:

1. $12\sqrt{3} \approx 20.7846$

2. $\sqrt{324} = 18$

3. $10.4$ (between 10 and 11; $10.4^2 = 108.16$ and $10.3^2 = 106.09$, and 108 is closer to 108.16)

4. $\frac{12}{6\sqrt{3}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \approx 1.1547$

5. $\sqrt{4} = 2$

If question 4 felt awkward, return to the simplification section: simplifying $\sqrt{108}$ first makes the denominator small enough to rationalise in one step.

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