Square Root of 105 — Value and How to Find It

The square root of 105 is $\sqrt{105} \approx 10.247$, and it stays under the radical because 105 has no square factor.

Quick Answer:

Result: $\sqrt{105} \approx 10.247$

Notation: $\sqrt{105}$ or $105^{1/2}$

Method shown: long division and prime-factorization check

Approximate value: $10.247$ (to 3 decimal places, irrational)

Exact form: $\sqrt{105}$ (already in simplest radical form)

Quick Reference Table

Where the Square Root of 105 Appears

The square root of 105 is the side length of a square whose area is 105 square units. It also shows up as a diagonal: a rectangle with sides measuring roughly $\sqrt{5}$ and $\sqrt{21}$ has a diagonal of $\sqrt{5 + 100}$ style calculations, and any right triangle whose legs square-sum to 105 has a hypotenuse of exactly $\sqrt{105}$.

What Is the Square Root of 105?

The square root of a number is the value that, multiplied by itself, gives that number. Since 105 is not a perfect square, it sits between $10^2 = 100$ and $11^2 = 121$, so its square root is between 10 and 11.

The result is an irrational number that never terminates and never repeats. Its prime factorization, $105 = 3 \times 5 \times 7$, has no repeated factor, so nothing can leave the radical, which is the same rule at work across squares and square roots.

How to Find the Square Root of 105 (Methods)

Method 1: Prime-factorization check

Break 105 into prime factors.

$$105 = 3 \times 5 \times 7$$

A factor leaves a square root only when it appears twice. Every prime here appears once.

$$\sqrt{105} = \sqrt{3 \times 5 \times 7}$$

Final answer: $\sqrt{105}$ does not simplify; it stays as $\sqrt{105}$.

Method 2: Long division for the decimal value

Pair the digits of 105 from the right: $1\,|\,05$. The largest square below 1 is $1^2 = 1$, so the first digit is 1.

$$1^2 = 1, \quad 1 - 1 = 0$$

Bring down 05 to get 5, and double the quotient to get 2 as the new divisor start. Find a digit $x$ with $2x \times x \le 500$ after adding decimals; $20 \times 0$ is too small, so extend to $102 \times ?$.

$$204 \times 2 = 408 \le 500$$ $$205 \times 5 = 1025 > 500$$

The next digit is 2, giving 10.2 so far. Continuing the process yields the next decimals.

$$\sqrt{105} \approx 10.247$$

Final answer: $\sqrt{105} \approx 10.247$ to three decimal places.

Common Mistakes With Square Root of 105

Mistake 1: Trying to pull a factor out

Where it slips in: assuming every large number simplifies like $\sqrt{108} = 6\sqrt{3}$.

Don't do this: writing $\sqrt{105}$ as some smaller radical times an integer.

The correct way: check the factorization first. Since $105 = 3 \times 5 \times 7$ has no repeated prime, $\sqrt{105}$ is already simplest.

Mistake 2: Rounding too early

Where it slips in: stopping the long division at 10.2 and calling it done.

Don't do this: writing $\sqrt{105} = 10.2$.

The correct way: carry the division to the required precision, $\approx 10.247$, before rounding.

Mistake 3: Confusing the square of 105 with its square root

Where it slips in: misreading the question.

Don't do this: answering $105^2 = 11025$.

The correct way: the square root asks for the number whose square is 105, which is $\approx 10.247$.

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