Square Root of 756

√756 at a glance

Exact value

6√21

Decimal (10 places)

27.4954541697

Rounded

27.5 · 27.50 · 27.495

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.495454

Prime factorization

2² × 3³ × 7

Cube root

9.109767

How to simplify √756

Look for the largest perfect square that divides 756. Here it is 36 (6²), because 756 = 36 × 21 and 21 has no square factor left:

The prime factorization tells the same story: 756 = 2² × 3³ × 7. Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 3 × 7 stays inside.

756 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √756 = 2√189, and √189 can be simplified again. Using 36 straight away finishes in one step.

Check: (6√21)² = 6² × 21 = 36 × 21 = 756. As a decimal, 6√21 = 6 × 4.582575695 ≈ 27.4954541697.

Where √756 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √756 lies between 27 and 28. 756 is 27 above 729 and 28 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.4909 (0.02% low)
• Tangent from 27, i.e. 27 + 27 ÷ 54: 27.5000 (0.02% high)
• Tangent from 28, i.e. 28 − 28 ÷ 56: 27.5000 (0.02% high)

For √756 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √756 with the Babylonian method

Picture a rectangle with an area of 756 and one side x; the other side must be 756 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √756.

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 √756 = 27.4954541697 to every decimal shown.

√756 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √756 the pattern is [27; 2, 54] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √756 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 − 756y² = 1. Its smallest solution in positive whole numbers is x = 55, y = 2.

√756 in geometry and everyday measurements

• 756 square feet is 70.2 m². Laid out as a square — a small house footprint or a lot — it is about 27.5 ft (27 ft 6 in) on a side.
• 756 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √756 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 8 × 26 box, because 4² + 8² + 26² = 756.
• Since √756 = 6√21, a length of √756 is exactly 6 copies of the length √21 laid end to end.

Square roots near √756 and related roots

• The cube root of 756 is about 9.109767.
• Because 756 = 4 × 189, the root is twice √189: 2 × 13.747727 ≈ 27.495454.

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