Square Root of 168 - Value, Simplest Form, and Examples

What Is A Square Root?

A square root of a number $n$ is a value $r$ such that $r^2 = n$. The square root of 168 is the positive number that, multiplied by itself, gives 168.

No whole number works, because $12^2 = 144$ is too small and $13^2 = 169$ is too big. So $\sqrt{168}$ lands between 12 and 13, and because 168 is only one less than the perfect square 169, the value sits very close to 13.

Where Does √168 Appear?

$\sqrt{168}$ is the space diagonal of a rectangular box measuring $2 \times 8 \times 10$ units, because the 3D diagonal equals $\sqrt{2^2 + 8^2 + 10^2} = \sqrt{4 + 64 + 100} = \sqrt{168}$. It also shows up whenever a problem scales $\sqrt{42}$ by 2, since $2\sqrt{42}$ and $\sqrt{168}$ are the same number written two ways.

Quick Reference Table

Is The Square Root Of 168 Rational Or Irrational?

$\sqrt{168}$ is irrational, meaning it cannot be written as a fraction $\frac{p}{q}$ of two integers, and its decimal never terminates or repeats.

Here is the quick reasoning. A whole number has a rational square root only when it is a perfect square — a number like 144, 169, or 196. Since 168 falls strictly between $12^2$ and $13^2$, it is not a perfect square, so its root is an irrational number.

Any decimal you write for $\sqrt{168}$, even with a hundred digits, is only an approximation. The exact value lives in the symbol $2\sqrt{42}$ itself. For a fuller treatment of why non-square integers behave this way, see Wikipedia's article on the square root.

How Do You Find √168? (Prime Factorization And Long Division)

Prime factorization (for the exact simplest form).

$$168 = 2 \times 2 \times 2 \times 3 \times 7$$

$$168 = (2 \times 2) \times 42$$

$$\sqrt{168} = \sqrt{4 \times 42}$$

$$\sqrt{168} = \sqrt{4} \times \sqrt{42}$$

$$\sqrt{168} = 2\sqrt{42}$$

Since $42 = 2 \times 3 \times 7$ carries no perfect-square factor, $2\sqrt{42}$ is fully simplified. This is exactly the process taught for simplifying radical expressions.

Long division (for the decimal value).

Step 1: Pair the digits around the decimal point: $\overline{1}\ \overline{68}.\overline{00}\ \overline{00}$.

Step 2: The largest square $\leq 1$ is $1$ ($1^2 = 1$). First quotient digit is $1$; remainder $0$.

Step 3: Bring down $68$ to get $68$. Double the quotient: $1 \to 2$. Find $d$ with $(20 + d),d \leq 68$; $d = 2$ gives $22 \times 2 = 44$. Quotient $12$, remainder $24$.

Step 4: Bring down $00$ to get $2400$. Double $12 \to 24$. Find $d$ with $(240 + d),d \leq 2400$; $d = 9$ gives $249 \times 9 = 2241$. Quotient $12.9$, remainder $159$.

Step 5: Bring down $00$ to get $15900$. Double $129 \to 258$. Find $d$ with $(2580 + d),d \leq 15900$; $d = 6$ gives $2586 \times 6 = 15516$. Quotient $12.96$, remainder $384$.

Step 6: Continue one more place to reach $\sqrt{168} \approx 12.9615$. The process never ends, which is what "irrational" means in practice.

Examples Of √168

Example 1

Simplify $\sqrt{168}$ to simplest radical form.

$$\sqrt{168} = \sqrt{4 \times 42}$$

$$\sqrt{168} = 2\sqrt{42}$$

Final answer: $2\sqrt{42}$.

Example 2

Between which two whole numbers does $\sqrt{168}$ lie? First instinct, then the check.

A tempting first move is to halve 168 and guess the root is near 84. Test it: $84^2 = 7056$, nowhere near 168. Halving finds a factor, not a root.

The correct approach compares 168 with nearby perfect squares.

$$12^2 = 144$$

$$13^2 = 169$$

Since $144 < 168 < 169$, the root lies between 12 and 13, and because 168 is just below 169, $\sqrt{168} \approx 12.96$.

Example 3

Evaluate $3\sqrt{168}$ in simplest form.

$$3\sqrt{168} = 3 \times 2\sqrt{42}$$

$$3\sqrt{168} = 6\sqrt{42}$$

Final answer: $6\sqrt{42} \approx 38.88$.

Example 4

Confirm that squaring the root returns 168.

$$(\sqrt{168})^2 = 168$$

$$(2\sqrt{42})^2 = 4 \times 42 = 168$$

Both forms agree, which is a fast way to verify a simplification.

Example 5

A box measures $2 \times 8 \times 10$ units. How long is its space diagonal?

$$d = \sqrt{2^2 + 8^2 + 10^2}$$

$$d = \sqrt{4 + 64 + 100}$$

$$d = \sqrt{168} = 2\sqrt{42} \approx 12.96 \text{ units}$$

Common Mistakes

Mistake 1: Pulling out the factor without square-rooting it

Where it slips in: Simplifying $\sqrt{168}$ after spotting the factor 4.

Don't do this: Writing $\sqrt{168} = \sqrt{4 \times 42} = 4\sqrt{42}$.

The correct way: The 4 leaves the radical as its own square root: $\sqrt{4} = 2$, so $\sqrt{168} = 2\sqrt{42}$. The rusher who moves the whole 4 out doubles the true answer, and a quick check ($4\sqrt{42} \approx 25.9$, far past 13) exposes the error.

Mistake 2: Treating 168 as a perfect square

Where it slips in: Because 168 sits right next to $169 = 13^2$.

Don't do this: Reporting $\sqrt{168} = 13$ or rounding it to a whole number and moving on.

The correct way: 168 is not a perfect square, so the root stays irrational: $2\sqrt{42} \approx 12.9615$. Closeness to 169 does not make 168 square.

Mistake 3: Rounding too early in a longer problem

Where it slips in: Multi-step questions where $\sqrt{168}$ appears mid-calculation.

Don't do this: Replace $\sqrt{168}$ with 12.96 at the start and carry that through every step.

The correct way: Keep the exact form $2\sqrt{42}$ until the final line, then round once. Early rounding compounds error across multiplications.

Conclusion

The square root of 168 is $2\sqrt{42}$, roughly $12.9615$, and it stays irrational because 168 is one short of the perfect square 169. Prime factorization gives the exact form; long division gives the decimal. To take radicals further with a teacher, explore Bhanzu's algebra tutor or a high school math tutor, and browse live math classes online. Ready to try one? Book a free demo class.

Read More

• Square Root 1 to 30 — every root from 1 to 30 in one reference table.

• Squares and Square Roots — how squaring and rooting undo each other.

• Square Root of 161 — the neighbouring non-square root, worked in full.

• Square Root of 10 — a smaller irrational root explained step by step.

• Square Root Tricks — fast estimation methods for non-perfect squares.