Square Root of 104 - 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 104 is the positive number that, multiplied by itself, gives 104.

No whole number works, because $10^2 = 100$ is too small and $11^2 = 121$ is too big. So $\sqrt{104}$ lands between 10 and 11, just above 10.2, since 104 is only a little more than 100.

Where Does √104 Appear?

$\sqrt{104}$ is the hypotenuse of a right triangle with legs 2 and 10, because the Pythagorean theorem gives $\sqrt{2^2 + 10^2} = \sqrt{4 + 100} = \sqrt{104}$. It also shows up as the distance between two grid points separated by 2 units across and 10 units up.

Quick Reference Table

Is The Square Root Of 104 Rational Or Irrational?

$\sqrt{104}$ is irrational - it cannot be written as a fraction $\frac{p}{q}$ of integers, and its decimal never terminates or repeats.

A whole number has a rational square root only when it is a perfect square. Since 104 falls strictly between $10^2$ and $11^2$, it is not a perfect square, so its root is an irrational number.

Simplifying does not remove the irrationality; it only tidies the form. Since $104 = 4 \times 26$, one perfect square (the 4) comes out, leaving $2\sqrt{26}$, and 26 carries no further square. For the formal definition of the square-root operation, see Wolfram MathWorld on the square root.

How Do You Find √104? (Prime Factorization And Long Division)

Prime factorization (for the exact simplest form).

$$104 = 2 \times 2 \times 2 \times 13$$

$$104 = (2 \times 2) \times 26$$

$$\sqrt{104} = \sqrt{4 \times 26}$$

$$\sqrt{104} = \sqrt{4} \times \sqrt{26}$$

$$\sqrt{104} = 2\sqrt{26}$$

Because $26 = 2 \times 13$ has no perfect-square factor, $2\sqrt{26}$ is fully simplified. This mirrors the process in simplifying radical expressions.

Long division (for the decimal value).

Step 1: Pair the digits around the decimal point: $\overline{1}\ \overline{04}.\overline{00}\ \overline{00}$.

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

Step 3: Bring down $04$ to get $4$. Double the quotient: $1 \to 2$. Find $d$ with $(20 + d),d \leq 4$; $d = 0$ works. Quotient $10$, remainder $4$.

Step 4: Bring down $00$ to get $400$. Double $10 \to 20$. Find $d$ with $(200 + d),d \leq 400$; $d = 1$ gives $201 \times 1 = 201$. Quotient $10.1$, remainder $199$.

Step 5: Bring down $00$ to get $19900$. Double $101 \to 202$. Find $d$ with $(2020 + d),d \leq 19900$; $d = 9$ gives $2029 \times 9 = 18261$. Quotient $10.19$, remainder $1639$.

Step 6: Continue one more place to reach $\sqrt{104} \approx 10.1980$. Estimation shortcuts for roots like this live in square root tricks.

Examples Of √104

Example 1

Simplify $\sqrt{104}$ to simplest radical form.

$$\sqrt{104} = \sqrt{4 \times 26}$$

$$\sqrt{104} = 2\sqrt{26}$$

Final answer: $2\sqrt{26}$.

Example 2

Simplify $\sqrt{104}$ after spotting the factor 4. First instinct, then the check.

A tempting first move is $\sqrt{104} = \sqrt{4 \times 26} = 4\sqrt{26}$, moving the whole 4 outside. Test it: $4\sqrt{26} \approx 4 \times 5.1 = 20.4$, yet the root must be near 10.2. The answer came out doubled.

The break is that the 4 leaves as its square root, not as itself.

$$\sqrt{4} = 2$$

$$\sqrt{104} = 2\sqrt{26} \approx 10.2$$

Example 3

Confirm that squaring the simplified form returns 104.

$$(2\sqrt{26})^2 = 2^2 \times (\sqrt{26})^2$$

$$(2\sqrt{26})^2 = 4 \times 26 = 104$$

Example 4

Evaluate $3\sqrt{104}$ in simplest form.

$$3\sqrt{104} = 3 \times 2\sqrt{26}$$

$$3\sqrt{104} = 6\sqrt{26} \approx 30.59$$

Example 5

A right triangle has legs 2 and 10. How long is its hypotenuse?

$$h = \sqrt{2^2 + 10^2}$$

$$h = \sqrt{4 + 100}$$

$$h = \sqrt{104} = 2\sqrt{26} \approx 10.2 \text{ units}$$

Common Mistakes

Mistake 1: Splitting the radical over addition

Where it slips in: Rewriting 104 as $100 + 4$ and rooting each piece.

Don't do this: Writing $\sqrt{104} = \sqrt{100} + \sqrt{4} = 10 + 2 = 12$.

The correct way: $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$, and $12^2 = 144 \neq 104$. Estimate against neighbouring squares instead, which keeps $\sqrt{104}$ between 10 and 11. The second-guesser who tries this and then distrusts the clean answer usually just needs the square-check to confirm 12 is too big.

Mistake 2: Treating 104 as almost-a-perfect-square

Where it slips in: Because 104 is close to $100 = 10^2$.

Don't do this: Rounding $\sqrt{104}$ down to a flat 10.

The correct way: 104 is not a perfect square, so its root is irrational, $2\sqrt{26} \approx 10.1980$. Nearness to 100 does not make 104 square.

Mistake 3: Rounding too early

Where it slips in: Longer problems where $\sqrt{104}$ appears partway through.

Don't do this: Replace $\sqrt{104}$ with 10.2 at the start and carry that value onward.

The correct way: Keep the exact $2\sqrt{26}$ until the final step, then round once, so error does not compound.

Conclusion

The square root of 104 is $2\sqrt{26}$, roughly $10.1980$, and it stays irrational because 104 is not a perfect square. Prime factorization pulls the 4 out as a 2; long division supplies the decimal; the squares 100 and 121 pin down its size. To build radical fluency with a teacher, explore Bhanzu's algebra tutor or a high school math tutor, or browse math classes online. Ready to see a lesson? 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 reverse each other.

• Square Root of 109 — the adjacent prime root that does not simplify.

• Square Root of 105 — a close non-square neighbour worked in full.

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