Square Root of 126 - Simplified Form, Value, and Example

What Is A Square Root?

The square root of a number $n$ is a value $r$ such that $r^2 = n$. The square root of 126 is the number that, multiplied by itself, gives 126.

No integer does this, because $11^2 = 121$ (too small) and $12^2 = 144$ (too big). So $\sqrt{126}$ lies between 11 and 12, close to 11.2.

The number under the radical sign - the radicand - is 126. Simplifying means rewriting it as a perfect square times a leftover, then taking the root of the perfect square.

Where Does √126 Appear?

$\sqrt{126}$ is the space diagonal of a $3 \times 6 \times 9$ box - the 3D distance rule gives $\sqrt{3^2 + 6^2 + 9^2} = \sqrt{9 + 36 + 81} = \sqrt{126}$. So a rectangular box with those edge lengths has a corner-to-opposite-corner reach of exactly $\sqrt{126}$ units. The value also appears whenever an algebra problem needs the largest perfect-square factor pulled from a radical, since $126 = 9 \times 14$ is a clean example of that extraction.

Quick Reference Table

Is The Square Root Of 126 Rational Or Irrational?

$\sqrt{126}$ is irrational - it cannot be written as a fraction $\frac{p}{q}$ of two integers, and its decimal neither terminates nor repeats.

The quick test. A whole number has a rational square root only when it is a perfect square. 126 is not a perfect square, so $\sqrt{126}$ is irrational.

The prime-factor reason. Factor the radicand:

$$126 = 2 \times 3^2 \times 7$$

For a square root to be rational, every prime must appear an even number of times. Here 2 and 7 each appear once - odd powers - so the root cannot resolve to a whole number or a fraction. After simplification, those odd primes stay trapped inside $\sqrt{14}$.

How Do You Find √126?

Prime factorization gives the exact simplified form; long division gives the decimal.

Prime Factorization (Exact Form)

Break 126 into primes and pair off the squares:

$$126 = 2 \times 3^2 \times 7$$

$$\sqrt{126} = \sqrt{3^2 \times 14}$$

$$\sqrt{126} = \sqrt{3^2} \times \sqrt{14}$$

$$\sqrt{126} = 3\sqrt{14}$$

The largest perfect-square factor of 126 is $9 = 3^2$, which comes out as 3. What remains, $\sqrt{14}$, has no perfect-square factor, so $3\sqrt{14}$ is the simplest radical form.

Long Division (Decimal Value)

Step 1: Pair the digits from the decimal point: $\overline{1},\overline{26}.\overline{00},\overline{00}$.

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

Step 3: Bring down 26 to make 26. Double the quotient: $1 \times 2 = 2$. Find $d$ with $(20 + d),d \leq 26$: $d = 1$ gives $21 \times 1 = 21$. Quotient 11; remainder 5.

Step 4: Add the decimal point, bring down 00 to make 500. Double 11 to get 22. Find $d$ with $(220 + d),d \leq 500$: $d = 2$ gives $222 \times 2 = 444$. Quotient 11.2; remainder 56.

Step 5: Bring down 00 to make 5600. Double 112 to get 224. Find $d$ with $(2240 + d),d \leq 5600$: $d = 2$ gives $2242 \times 2 = 4484$. Quotient 11.22; remainder 1116.

Step 6: Bring down 00 to make 111600. Double 1122 to get 2244. Find $d$ with $(22440 + d),d \leq 111600$: $d = 4$ gives $22444 \times 4 = 89776$. Quotient 11.224; remainder 21824.

Continuing gives $\sqrt{126} \approx 11.224$, and to four decimals $\sqrt{126} \approx 11.2250$.

Examples Of √126

Example 1: Confirm the simplified form

Show that $3\sqrt{14}$ squares back to 126.

$$\left(3\sqrt{14}\right)^2 = 3^2 \times \left(\sqrt{14}\right)^2$$

$$= 9 \times 14$$

$$= 126$$

Final answer: $\left(3\sqrt{14}\right)^2 = 126$, so the simplification checks out.

Example 2: Where students lose the mark

Simplify $\sqrt{126}$.

The tempting path: A student writes $\sqrt{126} = \sqrt{2 \times 63} = 2\sqrt{63}$, treating the 2 as if it came out.

Where it breaks: Only a perfect square leaves the radical, and 2 is not a perfect square - it should have stayed inside. Students first meeting prime factorization often mistake any factor for one that can be extracted. A check exposes it: $\left(2\sqrt{63}\right)^2 = 4 \times 63 = 252 \neq 126$.

The rescue: Pull out only the squared prime, $3^2$:

$$\sqrt{126} = \sqrt{9 \times 14} = 3\sqrt{14}$$

Final answer: $\sqrt{126} = 3\sqrt{14}$.

Example 3: Multiply two radicals

Simplify $\sqrt{126} \times \sqrt{14}$.

$$\sqrt{126} \times \sqrt{14} = \sqrt{126 \times 14}$$

$$= \sqrt{1764}$$

$$= 42$$

Final answer: $\sqrt{126} \times \sqrt{14} = 42$, because $3\sqrt{14} \times \sqrt{14} = 3 \times 14 = 42$.

Example 4: Estimate between perfect squares

Estimate $\sqrt{126}$ to one decimal place.

Since $11^2 = 121$ and $12^2 = 144$, the root lies between 11 and 12. The gap $126 - 121 = 5$ out of the interval width $144 - 121 = 23$ gives roughly $11 + \frac{5}{23} \approx 11.2$.

Final answer: $\sqrt{126} \approx 11.2$, close to the true $11.2250$.

Common Mistakes

Mistake 1: Pulling out a non-square factor

Where it slips in: Reading the prime factorization too quickly.

Don't do this: Writing $\sqrt{126} = 2\sqrt{63}$, as if the 2 leaves the radical.

The correct way: Only perfect squares come out. The squared prime here is $3^2 = 9$, giving $3\sqrt{14}$.

Mistake 2: Calling 126 a perfect square

Where it slips in: Rushing the rational-or-irrational check.

Don't do this: Expecting $\sqrt{126}$ to be a whole number.

The correct way: 126 sits between $121 = 11^2$ and $144 = 12^2$, so its root is irrational.

Mistake 3: Rounding too early

Where it slips in: Replacing $\sqrt{126}$ with 11.22 at the start of a longer calculation.

Don't do this: Carrying a rounded 11.22 through every step.

The correct way: Keep the exact form $3\sqrt{14}$ until the final line, then round once.

Conclusion

The square root of 126 is $3\sqrt{14} \approx 11.2250$: not a perfect square, irrational, and simplified by extracting the perfect-square factor 9 to leave $\sqrt{14}$ under the radical. Prime factorization delivers the exact form, and long division delivers the decimal. To go deeper into radicals with a teacher, explore Bhanzu's algebra tutor sessions, get targeted help with algebra, or join structured math classes online.

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Read More

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

• Simplifying Radical Expressions — the full method for reducing any radical to simplest form.

• Square Root Tricks — faster ways to estimate roots between perfect squares.

• Squares and Square Roots — the core rules linking squaring and its inverse.

• Square Root of 115 — a neighbouring non-perfect square that stays fully under the radical.

• Prove That Root 7 Is Irrational — the contradiction proof behind the $\sqrt{14}$ in $3\sqrt{14}$.