Square Root of 84

√84 at a glance

Exact value

2√21

Decimal (10 places)

9.1651513899

Rounded

9.2 · 9.17 · 9.165

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.165151

Prime factorization

2² × 3 × 7

Cube root

4.379519

How to simplify √84

Look for the largest perfect square that divides 84. Here it is 4 (2²), because 84 = 4 × 21 and 21 has no square factor left:

The prime factorization tells the same story: 84 = 2² × 3 × 7. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 7 stays inside.

Check: (2√21)² = 2² × 21 = 4 × 21 = 84. As a decimal, 2√21 = 2 × 4.582575695 ≈ 9.1651513899.

Where √84 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √84 lies between 9 and 10. 84 is 3 above 81 and 16 below 100, so the root is closer to 9.

• Straight line between 81 and 100: 9.1579 (0.08% low)
• Tangent from 9, i.e. 9 + 3 ÷ 18: 9.1667 (0.02% high)
• Tangent from 10, i.e. 10 − 16 ÷ 20: 9.2000 (0.38% high)

For √84 the tangent at 9 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 84 is just 3 above 81.

Finding √84 with the Babylonian method

Picture a rectangle with an area of 84 and one side x; the other side must be 84 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √84.

Start from the nearest whole number, 9 (9² = 81):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √84 = 9.1651513899 to every decimal shown.

√84 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √84 the pattern is [9; 6, 18] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √84 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 − 84y² = 1. Its smallest solution in positive whole numbers is x = 55, y = 6.

√84 in geometry and everyday measurements

• A square room or garden bed covering 84 square feet measures about 9.17 ft (9 ft 2 in) along each wall.
• 84 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 √84 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 8 box, because 2² + 4² + 8² = 84.
• Since √84 = 2√21, a length of √84 is exactly 2 copies of the length √21 laid end to end.

Square roots near √84 and related roots

• The cube root of 84 is about 4.379519.
• Four times the radicand doubles the root: √336 = 2 × √84 ≈ 18.330303.

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