The square root of 157 is $\sqrt{157} \approx 12.530$, and because 157 is prime, it never leaves the radical.
Quick Answer:
Result: $\sqrt{157} \approx 12.530$
Notation: $\sqrt{157}$ or $157^{1/2}$
Method shown: long division and prime check
Approximate value: $12.530$ (to 3 decimal places, irrational)
Exact form: $\sqrt{157}$ (already in simplest radical form)
Quick Reference Table
Number | Simplified square root | Decimal (3 dp) |
|---|---|---|
$\sqrt{144}$ | $12$ | $12.000$ |
$\sqrt{150}$ | $5\sqrt{6}$ | $12.247$ |
$\sqrt{156}$ | $2\sqrt{39}$ | $12.490$ |
$\sqrt{157}$ | $\sqrt{157}$ | $12.530$ |
$\sqrt{160}$ | $4\sqrt{10}$ | $12.649$ |
$\sqrt{169}$ | $13$ | $13.000$ |
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 $6^2 + 11^2 = 36 + 121 = 157$, so its longest side measures exactly $\sqrt{157} \approx 12.530$.
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 $12^2 = 144$ and $13^2 = 169$, 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 $\sqrt{157}$ 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.
$$157 = 157$$
Since 157 is prime, it has no repeated factor, and no factor appears twice to leave the radical.
$$\sqrt{157} = \sqrt{157}$$
Final answer: $\sqrt{157}$ does not simplify.
Method 2: Long division for the decimal value
Pair the digits of 157 from the right: $1\,|\,57$. The largest square at or below 1 is $1^2 = 1$, so the first digit is 1.
$$1^2 = 1, \quad 1 - 1 = 0$$
Bring down 57 to get 57, and double the quotient to start the divisor at 2. Find a digit $x$ with $2x \times x \le 57$.
$$22 \times 2 = 44 \le 57$$ $$23 \times 3 = 69 > 57$$
The next digit is 2, giving 12 with remainder 13. Adding decimal places and continuing gives the next digits.
$$\sqrt{157} \approx 12.530$$
Final answer: $\sqrt{157} \approx 12.530$ 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 $\sqrt{160} = 4\sqrt{10}$.
Don't do this: writing $\sqrt{157}$ as an integer times a smaller radical.
The correct way: 157 is prime, so no factor appears twice and $\sqrt{157}$ 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 $12.5^2 = 156.25$, the root is just above 12.5, giving $\approx 12.530$.
Mistake 3: Ignoring the negative root
Where it slips in: solving $x^2 = 157$ and giving only the positive value.
Don't do this: writing $x = 12.530$ only.
The correct way: an equation like $x^2 = 157$ has two solutions, $x \approx \pm 12.530$.
To build square-root fluency with a teacher, explore Bhanzu's algebra tutor or math classes online.
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