Square Root of 147

√147 at a glance

Exact value

7√3

Decimal (10 places)

12.1243556530

Rounded

12.1 · 12.12 · 12.124

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.124356

Prime factorization

3 × 7²

Cube root

5.277632

How to simplify √147

Look for the largest perfect square that divides 147. Here it is 49 (7²), because 147 = 49 × 3 and 3 has no square factor left:

The prime factorization tells the same story: 147 = 3 × 7². Each pair of equal primes leaves the radical as one factor, so 7 comes out and 3 stays inside.

Check: (7√3)² = 7² × 3 = 49 × 3 = 147. As a decimal, 7√3 = 7 × 1.7320508076 ≈ 12.1243556530.

Where √147 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √147 lies between 12 and 13. 147 is 3 above 144 and 22 below 169, so the root is closer to 12.

• Straight line between 144 and 169: 12.1200 (0.04% low)
• Tangent from 12, i.e. 12 + 3 ÷ 24: 12.1250 (0.01% high)
• Tangent from 13, i.e. 13 − 22 ÷ 26: 12.1538 (0.24% high)

For √147 the tangent at 12 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 147 is just 3 above 144.

Finding √147 with the Babylonian method

If a guess is too big, 147 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√147) in one step.

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

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √147 = 12.1243556530 to every decimal shown.

√147 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √147 the pattern is [12; 8, 24] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √147 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 − 147y² = 1. Its smallest solution in positive whole numbers is x = 97, y = 8.

√147 in geometry and everyday measurements

• A square room or garden bed covering 147 square feet measures about 12.12 ft (12 ft 1 in) along each wall.
• 147 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 √147 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 11 box, because 1² + 5² + 11² = 147.
• Since √147 = 7√3, a length of √147 is exactly 7 copies of the length √3 laid end to end.

Square roots near √147 and related roots

• The cube root of 147 is about 5.277632.
• Four times the radicand doubles the root: √588 = 2 × √147 ≈ 24.248711.

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