Square Root of 977

√977 at a glance

Exact value

√977

Decimal (10 places)

31.2569992162

Rounded

31.3 · 31.26 · 31.257

Perfect square?

No — between 31² and 32²

Rational?

Irrational

Both square roots

±31.256999

Prime factorization

977

Cube root

9.922738

How to simplify √977

977 is a prime number, so its only factors are 1 and 977. There is no perfect-square factor to pull out, which means √977 is already in its simplest radical form.

The square root of any prime is irrational. If √977 were a fraction a/b in lowest terms, then a² = 977b², so 977 would divide a — and then 977 would divide b too, contradicting “lowest terms.” That is why the decimal 31.2569992162 is only a rounded value.

Where √977 sits between perfect squares

961 = 31² and 1,024 = 32² are the nearest perfect squares, so √977 lies between 31 and 32. 977 is 16 above 961 and 47 below 1,024, so the root is closer to 31.

• Straight line between 961 and 1,024: 31.2540 (0.01% low)
• Tangent from 31, i.e. 31 + 16 ÷ 62: 31.2581 (0% high)
• Tangent from 32, i.e. 32 − 47 ÷ 64: 31.2656 (0.03% high)

For √977 the tangent at 31 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 977 is just 16 above 961.

Finding √977 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 977: following the tangent line down to zero simplifies to averaging x with 977 ÷ x.

Start from the nearest whole number, 31 (31² = 961):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √977 = 31.2569992162 to every decimal shown.

√977 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √977 the pattern is [31; 3, 1, 8, 5, 1, 1, 3, 7, 1, 1, 7, 3, …] with the block of 19 terms after the semicolon repeating forever (only the first 12 of the 19 are shown). A pattern that never ends is one more proof that √977 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 977y² = 1. Its smallest solution in positive whole numbers is x = 108,832,847,723,078,562,849, y = 3,481,871,275,306,470,280 — 21 digits for x, even though 977 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 7,376,748,868² − 977 × 236,003,105² = −1.

√977 in geometry and everyday measurements

• 977 square feet is 90.8 m². Laid out as a square — a small house footprint or a lot — it is about 31.26 ft (31 ft 3 in) on a side.
• 977 = 4² + 31², so by the Pythagorean theorem √977 is the diagonal of a 4 × 31 rectangle — and the distance between the points (0, 0) and (4, 31) on a grid.

Square roots near √977 and related roots

• The cube root of 977 is about 9.922738.
• Squaring undoes the root: (√977)² = 977, while 977² = 954,529 — the number whose square root is 977.

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