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 137 is the number that, multiplied by itself, gives 137.
No whole number does it. $11^2 = 121$ is too small and $12^2 = 144$ is too big, so $\sqrt{137}$ lies between 11 and 12. Each positive number has a positive and a negative root, but $\sqrt{137}$ denotes the positive one.
Where Does √137 Appear?
$\sqrt{137}$ is the hypotenuse of a right triangle with legs 4 and 11. The Pythagorean theorem gives the hypotenuse as $\sqrt{4^2 + 11^2} = \sqrt{16 + 121} = \sqrt{137}$. Because 137 falls between the perfect squares $121 = 11^2$ and $144 = 12^2$, the value sits between 11 and 12, closer to 12. Any right triangle whose two legs square to a total of 137 has this exact length across from the right angle.
Quick Reference Table
Number $n$ | $\sqrt{n}$ (approx.) | Rational or Irrational |
|---|---|---|
121 | 11 | Rational |
130 | 11.4018 | Irrational |
137 | 11.7047 | Irrational |
140 | 11.8322 | Irrational |
144 | 12 | Rational |
150 | 12.2474 | Irrational |
169 | 13 | Rational |
Is The Square Root Of 137 Rational Or Irrational?
$\sqrt{137}$ is an irrational number. It cannot be written as a ratio of two integers, and its decimal never terminates or repeats.
The reason is quick for 137: it is a prime number, divisible only by 1 and itself. A prime has no repeated factor, so it can never be a perfect square (a perfect square needs every prime factor paired up). With nothing to pair and nothing to pull outside the radical, $\sqrt{137}$ stays exactly as it is and cannot be rational. This mirrors the classic proofs that the square root of a prime is always irrational.
How Do You Find √137? (Long Division Method)
Since 137 is prime, there is no factor to simplify, so the way to get its value is long division. Work one step per line.
Step 1: Write 137 as $\overline{1},\overline{37}.\overline{00},\overline{00}$, pairing digits from the decimal point.
Step 2: Find the largest number whose square is at most 1 (the first group). $1^2 = 1$ Write 1 as the first digit, remainder $0$.
Step 3: Bring down the next pair, 37, to get 37. Double the current answer, 1, to get 2. Find a digit $d$ with $(20 + d) \times d \leq 37$. $21 \times 1 = 21 \leq 37$ The answer is now 11.
Step 4: Subtract and bring down. $37 - 21 = 16$ Bring down $00$ to get $1600$.
Step 5: Double 11 to get 22. Find $d$ with $(220 + d) \times d \leq 1600$. $227 \times 7 = 1589 \leq 1600$ The answer is now $11.7$.
Step 6: Subtract and bring down. $1600 - 1589 = 11$ Bring down $00$ to get $1100$ $2340 \times 0 = 0$, so the next digit is 0, and the answer is $11.70$.
Carrying two more places gives $\sqrt{137} \approx 11.7047$. The decimal keeps going, which is the signature of an irrational number.
Examples Of √137
Example 1
Estimate $\sqrt{137}$ to the nearest whole number.
$11^2 = 121$ $12^2 = 144$ 137 is between 121 and 144, and closer to 144. Final answer: 12.
Example 2
A student writes $\sqrt{137} = \sqrt{137}$ but insists it should reduce to something like $\sqrt{100} + \sqrt{37}$. What breaks?
The wrong path first. Splitting a radical over addition is not allowed. Test it with easy numbers: $\sqrt{9 + 16} = \sqrt{25} = 5$, but $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$. Since $5 \neq 7$, the rule $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$ is false. Radicals only split over multiplication, and 137 is prime, so no split helps. Final answer: $\sqrt{137}$ is already in simplest form.
Example 3
Evaluate $(\sqrt{137})^2$.
Squaring undoes the square root. $(\sqrt{137})^2 = 137$ Final answer: 137.
Example 4
Between which two consecutive integers does $\sqrt{137}$ lie?
$11^2 = 121 < 137$ $12^2 = 144 > 137$ So $11 < \sqrt{137} < 12$. Final answer: between 11 and 12.
Example 5
Find $\sqrt{137}$ to two decimal places, then check by squaring.
From long division, $\sqrt{137} \approx 11.70$. Check: $11.70^2 = 136.89$, close to 137. Refining gives $11.7047^2 \approx 137.00$. Final answer: $\sqrt{137} \approx 11.70$.
Common Mistakes
Mistake 1: Splitting the radical over addition
Where it slips in: Trying to break 137 into two friendlier squares.
Don't do this: Writing $\sqrt{137} = \sqrt{121} + \sqrt{16} = 11 + 4 = 15$.
The correct way: $\sqrt{a + b}$ is not $\sqrt{a} + \sqrt{b}$. Since $15^2 = 225 \neq 137$, the split is wrong. Radicals distribute only over multiplication.
Mistake 2: Calling 137 a perfect square because it is close to 144
Where it slips in: Estimating and then treating the estimate as exact.
Don't do this: Rounding $\sqrt{137}$ to 12 and writing $\sqrt{137} = 12$.
The correct way: 12 is the nearest whole number, but $12^2 = 144$, not 137. Keep $\sqrt{137}$ exact and only round at the end.
Mistake 3: Trying to simplify a prime radical
Where it slips in: Assuming every root reduces.
Don't do this: Searching for a perfect-square factor of 137.
The correct way: 137 is prime, so its only factors are 1 and 137. There is no perfect-square factor above 1, and $\sqrt{137}$ is its own simplest form.
Conclusion
The square root of 137 is an irrational number, about $11.7047$, and $\sqrt{137}$ is already its simplest form because 137 is prime. There is no perfect-square factor to extract, so long division is the tool that delivers the decimal to any precision. To go deeper with a teacher, explore Bhanzu's algebra classes or work with a high school math tutor. You can also book a free demo class to see the method taught live.
Read More
Square root tricks for estimating roots quickly
Square root 1 to 30 as a reference chart
Square root of 157, another prime radical
Square root of 105, a nearby irrational value
Square root of 93, an irrational root that will not reduce
Square root of 365 for a similar computation
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