Square Root of 752

√752 at a glance

Exact value

4√47

Decimal (10 places)

27.4226184016

Rounded

27.4 · 27.42 · 27.423

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.422618

Prime factorization

2⁴ × 47

Cube root

9.093672

How to simplify √752

Look for the largest perfect square that divides 752. Here it is 16 (4²), because 752 = 16 × 47 and 47 has no square factor left:

The prime factorization tells the same story: 752 = 2⁴ × 47. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 47 stays inside.

752 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √752 = 2√188, and √188 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√47)² = 4² × 47 = 16 × 47 = 752. As a decimal, 4√47 = 4 × 6.8556546004 ≈ 27.4226184016.

Where √752 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √752 lies between 27 and 28. 752 is 23 above 729 and 32 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.4182 (0.02% low)
• Tangent from 27, i.e. 27 + 23 ÷ 54: 27.4259 (0.01% high)
• Tangent from 28, i.e. 28 − 32 ÷ 56: 27.4286 (0.02% high)

For √752 the tangent at 27 wins, missing by only 0.0033. Tangent estimates shine when the number sits close to a perfect square — here 752 is just 23 above 729.

Finding √752 with the Babylonian method

Picture a rectangle with an area of 752 and one side x; the other side must be 752 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √752.

Start from the nearest whole number, 27 (27² = 729):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √752 = 27.4226184016 to every decimal shown.

√752 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √752 the pattern is [27; 2, 2, 1, 2, 1, 2, 2, 54] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √752 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 − 752y² = 1. Its smallest solution in positive whole numbers is x = 4,607, y = 168.

√752 in geometry and everyday measurements

• 752 square feet is 69.9 m². Laid out as a square — a small house footprint or a lot — it is about 27.42 ft (27 ft 5 in) on a side.
• 752 is not a sum of two whole-number squares — the prime factor 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √752 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 √752 as its space diagonal.
• Since √752 = 4√47, a length of √752 is exactly 4 copies of the length √47 laid end to end.

Square roots near √752 and related roots

• The cube root of 752 is about 9.093672.
• Because 752 = 4 × 188, the root is twice √188: 2 × 13.711309 ≈ 27.422618.

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