What Is a Square Root?
The square root of a number $n$ is the value $r$ for which $r^2 = n$. The square root of 1681 is the number that, multiplied by itself, gives 1681.
Here an integer works exactly. Because $41^2 = 1681$, the root is the whole number 41, with no decimal tail.
Every positive number has two square roots, $41$ and $-41$, since $(-41)^2 = 1681$ as well. The symbol $\sqrt{1681}$ denotes the principal (positive) root, 41.
Where Does √1681 Appear?
$\sqrt{1681} = 41$ is the side length of a square whose area is 1681 square units, so a $41 \times 41$ grid holds exactly 1681 cells. The number also appears through the Pythagorean theorem: legs of 9 and 40 give a hypotenuse of 41, since $81 + 1600 = 1681$.
Quick Reference Table
Number $n$ | $\sqrt{n}$ | Perfect square? | Rational or Irrational |
|---|---|---|---|
1600 | 40 | Yes | Rational |
1650 | 40.6202 | No | Irrational |
1681 | 41 | Yes | Rational |
1700 | 41.2311 | No | Irrational |
1764 | 42 | Yes | Rational |
1849 | 43 | Yes | Rational |
Is the Square Root of 1681 Rational or Irrational?
$\sqrt{1681}$ is rational - in fact it is the integer 41, which can be written as $\frac{41}{1}$.
Why? A whole number has a rational square root exactly when it is a perfect square, and 1681 is a perfect square. So $\sqrt{1681}$ is rational, unlike the roots of nearby non-squares such as 1680 or 1682.
Is 1681 a perfect square even though 41 is prime? Yes. A perfect square is any integer times itself, and $41 \times 41$ qualifies - a point the square root 1 to 30 hub reinforces with smaller prime squares like $4 = 2^2$ and $9 = 3^2$.
How Do You Find √1681? (Prime Factorization and Long Division)
Prime factorization is short here because 41 is prime.
$$1681 = 41 \times 41$$
$$1681 = 41^2$$
The single pair of 41s means one 41 leaves the radical.
$$\sqrt{1681} = 41$$
Every prime pairs off with nothing left inside, the hallmark of a perfect square covered in squares and square roots.
Long division confirms the answer. Pair the digits: $16,81$.
Step 1: Find the largest integer whose square is $\leq 16$.
$$4^2 = 16 \leq 16$$
Step 2: Subtract and bring down the next pair.
$$16 - 16 = 0$$
$$\text{new dividend} = 81$$
Step 3: Double the quotient 4 to get 8, then find a digit $d$ with $(80 + d) \times d \leq 81$.
$$81 \times 1 = 81$$
Step 4: Subtract.
$$81 - 81 = 0$$
The remainder is zero, so the division ends exactly.
$$\sqrt{1681} = 41$$
Because 1681 is a clean perfect square, the last-digit shortcut in square root tricks also points to 41: a square ending in 1 has a root ending in 1 or 9, and only 41 fits between 40 and 42.
Examples of √1681
Example 1
Verify that $\sqrt{1681} = 41$.
$$41 \times 41 = 1681$$
The product returns 1681, so 41 is confirmed.
Final answer: $\sqrt{1681} = 41$.
Example 2
Decide whether 1681 has a whole-number square root.
The first instinct is to test small primes and stop early. Dividing by 7, 11, 13, 17, 23, and 37 leaves a remainder each time, so a student might conclude:
$$1681 \text{ is prime, so } \sqrt{1681} \text{ is irrational} \quad ?$$
Check further. Testing must continue up to $\sqrt{1681}$, and $1681 \div 41 = 41$, so 1681 is not prime at all.
The slip is stopping the divisibility test too soon. Since $1681 = 41^2$, it is a perfect square, and its root is the whole number 41.
Final answer: $\sqrt{1681} = 41$.
Example 3
Between which two consecutive multiples of ten does $\sqrt{1681}$ sit, and what is the exact value?
$$40^2 = 1600$$
$$42^2 = 1764$$
Since $1600 < 1681 < 1764$, the root is between 40 and 42, and testing $41^2 = 1681$ pins it exactly at 41.
Final answer: exactly 41.
Example 4
A square mosaic uses 1681 identical tiles arranged in a square. How many tiles line each side?
Tiles per side $= \sqrt{1681}$.
$$\sqrt{1681} = 41$$
Final answer: 41 tiles per side.
Example 5
Evaluate $\sqrt{1681} - \sqrt{1369}$.
Take each root first.
$$\sqrt{1681} = 41$$
$$\sqrt{1369} = 37$$
$$41 - 37 = 4$$
Final answer: 4.
Common Mistakes
Mistake 1: Stopping the divisibility test too soon
Where it slips in: When checking whether a large number like 1681 is prime or a perfect square.
Don't do this: Testing only up to 37, finding no divisor, and calling 1681 prime.
The correct way: Test divisors up to the square root of the number. Doing so reveals $1681 = 41 \times 41$, so it is the perfect square $41^2$. Halting the divisor search early is the habit that hides a perfect square behind a prime-looking number.
Mistake 2: Assuming a prime cannot produce a perfect square
Where it slips in: When a student learns 41 is prime.
Don't do this: Concluding that $\sqrt{1681}$ must be irrational because 41 is prime.
The correct way: A perfect square is any integer times itself, so $41^2 = 1681$ is a perfect square and its root is the whole number 41.
Mistake 3: Losing the negative root when solving an equation
Where it slips in: When solving $x^2 = 1681$.
Don't do this: Writing only $x = 41$.
The correct way: The equation $x^2 = 1681$ has two solutions, $x = 41$ and $x = -41$, written $x = \pm 41$. The symbol $\sqrt{1681}$ alone still means the positive root, 41.
Conclusion
The square root of 1681 is the whole number 41 because 1681 is the perfect square of the prime 41, confirmed by both prime factorization and long division. To build fluency with perfect squares and roots alongside a teacher, explore Bhanzu's algebra tutor or a high school math tutor, or join structured math classes online. Want the pairing method demonstrated live? Book a free demo class.
Read More
Square Root of 2025 — the perfect square 45, worked by the same methods.
Square Root of 441 — a perfect square equal to 21.
Square Root of 196 — the perfect square 14, with full working.
Square Root of 1369 — the neighbouring perfect square, equal to 37.
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