Square Root of 48

√48 at a glance

Exact value

4√3

Decimal (10 places)

6.9282032303

Rounded

6.9 · 6.93 · 6.928

Perfect square?

No — between 6² and 7²

Rational?

Irrational

Both square roots

±6.928203

Prime factorization

2⁴ × 3

Cube root

3.634241

How to simplify √48

Look for the largest perfect square that divides 48. Here it is 16 (4²), because 48 = 16 × 3 and 3 has no square factor left:

The prime factorization tells the same story: 48 = 2⁴ × 3. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 3 stays inside.

48 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √48 = 2√12, and √12 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√3)² = 4² × 3 = 16 × 3 = 48. As a decimal, 4√3 = 4 × 1.7320508076 ≈ 6.9282032303.

Where √48 sits between perfect squares

36 = 6² and 49 = 7² are the nearest perfect squares, so √48 lies between 6 and 7. 48 is 12 above 36 and 1 below 49, so the root is closer to 7.

• Straight line between 36 and 49: 6.9231 (0.07% low)
• Tangent from 6, i.e. 6 + 12 ÷ 12: 7.0000 (1.04% high)
• Tangent from 7, i.e. 7 − 1 ÷ 14: 6.9286 (0.01% high)

For √48 the tangent at 7 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 48 is just 1 below 49.

Finding √48 with the Babylonian method

Picture a rectangle with an area of 48 and one side x; the other side must be 48 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √48.

Start from the nearest whole number, 7 (7² = 49):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √48 = 6.9282032303 to every decimal shown.

√48 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √48 the pattern is [6; 1, 12] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √48 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 − 48y² = 1. Its smallest solution in positive whole numbers is x = 7, y = 1.

√48 in geometry and everyday measurements

• A square room or garden bed covering 48 square feet measures about 6.93 ft (6 ft 11 in) along each wall.
• 48 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √48 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 4 × 4 box, because 4² + 4² + 4² = 48.
• Since √48 = 4√3, a length of √48 is exactly 4 copies of the length √3 laid end to end.

Square roots near √48 and related roots

• The cube root of 48 is about 3.634241.
• Four times the radicand doubles the root: √192 = 2 × √48 ≈ 13.856406.

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