Square Root of 148

√148 at a glance

Exact value

2√37

Decimal (10 places)

12.1655250606

Rounded

12.2 · 12.17 · 12.166

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.165525

Prime factorization

2² × 37

Cube root

5.289572

How to simplify √148

Look for the largest perfect square that divides 148. Here it is 4 (2²), because 148 = 4 × 37 and 37 has no square factor left:

The prime factorization tells the same story: 148 = 2² × 37. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 37 stays inside.

Check: (2√37)² = 2² × 37 = 4 × 37 = 148. As a decimal, 2√37 = 2 × 6.0827625303 ≈ 12.1655250606.

Where √148 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √148 lies between 12 and 13. 148 is 4 above 144 and 21 below 169, so the root is closer to 12.

• Straight line between 144 and 169: 12.1600 (0.05% low)
• Tangent from 12, i.e. 12 + 4 ÷ 24: 12.1667 (0.01% high)
• Tangent from 13, i.e. 13 − 21 ÷ 26: 12.1923 (0.22% high)

For √148 the tangent at 12 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 148 is just 4 above 144.

Finding √148 with the Babylonian method

Picture a rectangle with an area of 148 and one side x; the other side must be 148 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √148.

Start from the nearest whole number, 12 (12² = 144):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √148 = 12.1655250606 to every decimal shown.

√148 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √148 the pattern is [12; 6, 24] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √148 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 − 148y² = 1. Its smallest solution in positive whole numbers is x = 73, y = 6.

√148 in geometry and everyday measurements

• A square room or garden bed covering 148 square feet measures about 12.17 ft (12 ft 2 in) along each wall.
• 148 = 2² + 12², so by the Pythagorean theorem √148 is the diagonal of a 2 × 12 rectangle — and the distance between the points (0, 0) and (2, 12) on a grid.
• Since √148 = 2√37, a length of √148 is exactly 2 copies of the length √37 laid end to end.

Square roots near √148 and related roots

• The cube root of 148 is about 5.289572.
• Four times the radicand doubles the root: √592 = 2 × √148 ≈ 24.33105.

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