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Square Root of 157 — Value and How to Find It

Algebra

TL;DR

The square root of 157 is 157≈12.530\sqrt{157} \approx 12.530157​≈12.530, an irrational number that cannot be simplified because 157 is prime. This article shows why the radical stays as is, the long-division method, where it appears, common mistakes, and worked examples.

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Bhanzu Team Last updated on July 20, 2026 4 min read

The square root of 157 is 157≈12.530\sqrt{157} \approx 12.530157​≈12.530, and because 157 is prime, it never leaves the radical.

Quick Answer:
Result: 157≈12.530\sqrt{157} \approx 12.530157​≈12.530
Notation: 157\sqrt{157}157​ or 1571/2157^{1/2}1571/2
Method shown: long division and prime check
Approximate value: 12.53012.53012.530 (to 3 decimal places, irrational)
Exact form: 157\sqrt{157}157​ (already in simplest radical form)

Quick Reference Table

Where the Square Root of 157 Appears

The square root of 157 is the side length of a square whose area is 157 square units. It also appears as a hypotenuse: a right triangle with legs 6 and 11 has 62+112=36+121=1576^2 + 11^2 = 36 + 121 = 15762+112=36+121=157, so its longest side measures exactly 157≈12.530\sqrt{157} \approx 12.530157​≈12.530.

What Is the Square Root of 157?

The square root of a number is the value that, multiplied by itself, gives that number. Since 157 is not a perfect square, it sits between 122=14412^2 = 144122=144 and 132=16913^2 = 169132=169, so its square root is between 12 and 13.

The result is an irrational number that never terminates and never repeats. Because 157 is prime, it has no factor pair to pull out, so 157\sqrt{157}157​ is already in simplest form, the same principle covered in squares and square roots.

How to Find the Square Root of 157 (Methods)

Method 1: Prime check

Test 157 for factors up to its square root, about 12.5. It is not divisible by 2, 3, 5, 7, or 11.

157=157157 = 157157=157

Since 157 is prime, it has no repeated factor, and no factor appears twice to leave the radical.

157=157\sqrt{157} = \sqrt{157}157​=157​

Final answer: 157\sqrt{157}157​ does not simplify.

Method 2: Long division for the decimal value

Pair the digits of 157 from the right: 1∣571, |571∣57. The largest square at or below 1 is 12=11^2 = 112=1, so the first digit is 1.

12=1,1−1=01^2 = 1, \quad 1 - 1 = 012=1,1−1=0

Bring down 57 to get 57, and double the quotient to start the divisor at 2. Find a digit xxx with 2x×x≤572x \times x \le 572x×x≤57.

22×2=44≤5722 \times 2 = 44 \le 5722×2=44≤5723×3=69>5723 \times 3 = 69 > 5723×3=69>57

The next digit is 2, giving 12 with remainder 13. Adding decimal places and continuing gives the next digits.

157≈12.530\sqrt{157} \approx 12.530157​≈12.530

Final answer: 157≈12.530\sqrt{157} \approx 12.530157​≈12.530 to three decimal places.

Common Mistakes With Square Root of 157

Mistake 1: Trying to simplify a prime radical

Where it slips in: assuming a three-digit number must break down like 160=410\sqrt{160} = 4\sqrt{10}160​=410​.

Don't do this: writing 157\sqrt{157}157​ as an integer times a smaller radical.

The correct way: 157 is prime, so no factor appears twice and 157\sqrt{157}157​ stays as is.

Mistake 2: Estimating too high

Where it slips in: seeing 157 is close to 169 and guessing the root is near 13.

Don't do this: answering roughly 12.9.

The correct way: since 12.52=156.2512.5^2 = 156.2512.52=156.25, the root is just above 12.5, giving ≈12.530\approx 12.530≈12.530.

Mistake 3: Ignoring the negative root

Where it slips in: solving x2=157x^2 = 157x2=157 and giving only the positive value.

Don't do this: writing x=12.530x = 12.530x=12.530 only.

The correct way: an equation like x2=157x^2 = 157x2=157 has two solutions, x≈±12.530x \approx \pm 12.530x≈±12.530.

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Frequently Asked Questions

What is the value of the square root of 157?

It is 157≈12.530\sqrt{157} \approx 12.530157​≈12.530, an irrational number.

Is the square root of 157 rational or irrational?

Irrational. Since 157 is not a perfect square, 157\sqrt{157}157​ is a non-terminating, non-repeating decimal.

Can 157\sqrt{157}157​ be simplified?

No. Because 157 is prime, it has no repeated factor, so nothing can leave the radical.

Is 157 a perfect square?

No. It lies between the perfect squares 144=122144 = 12^2144=122 and 169=132169 = 13^2169=132.

Between which two whole numbers does 157\sqrt{157}157​ lie?

Between 12 and 13, and closer to 13 since 157 is nearer to 169 than to 144.