Square Root of 988

√988 at a glance

Exact value

2√247

Decimal (10 places)

31.4324672910

Rounded

31.4 · 31.43 · 31.432

Perfect square?

No — between 31² and 32²

Rational?

Irrational

Both square roots

±31.432467

Prime factorization

2² × 13 × 19

Cube root

9.959839

How to simplify √988

Look for the largest perfect square that divides 988. Here it is 4 (2²), because 988 = 4 × 247 and 247 has no square factor left:

The prime factorization tells the same story: 988 = 2² × 13 × 19. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 13 × 19 stays inside.

Check: (2√247)² = 2² × 247 = 4 × 247 = 988. As a decimal, 2√247 = 2 × 15.7162336455 ≈ 31.4324672910.

Where √988 sits between perfect squares

961 = 31² and 1,024 = 32² are the nearest perfect squares, so √988 lies between 31 and 32. 988 is 27 above 961 and 36 below 1,024, so the root is closer to 31.

• Straight line between 961 and 1,024: 31.4286 (0.01% low)
• Tangent from 31, i.e. 31 + 27 ÷ 62: 31.4355 (0.01% high)
• Tangent from 32, i.e. 32 − 36 ÷ 64: 31.4375 (0.02% high)

For √988 the tangent at 31 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 988 is just 27 above 961.

Finding √988 with the Babylonian method

Picture a rectangle with an area of 988 and one side x; the other side must be 988 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √988.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √988 = 31.4324672910 to every decimal shown.

√988 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √988 the pattern is [31; 2, 3, 4, 1, 20, 6, 1, 14, 1, 6, 20, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √988 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 − 988y² = 1. Its smallest solution in positive whole numbers is x = 14,549,450,527, y = 462,879,684.

√988 in geometry and everyday measurements

• 988 square feet is 91.8 m². Laid out as a square — a small house footprint or a lot — it is about 31.43 ft (31 ft 5 in) on a side.
• 988 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √988 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √988 as its space diagonal.
• Since √988 = 2√247, a length of √988 is exactly 2 copies of the length √247 laid end to end.

Square roots near √988 and related roots

• The cube root of 988 is about 9.959839.
• Because 988 = 4 × 247, the root is twice √247: 2 × 15.716234 ≈ 31.432467.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.