Square Root of 700

√700 at a glance

Exact value

10√7

Decimal (10 places)

26.4575131106

Rounded

26.5 · 26.46 · 26.458

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.457513

Prime factorization

2² × 5² × 7

Cube root

8.879040

How to simplify √700

Look for the largest perfect square that divides 700. Here it is 100 (10²), because 700 = 100 × 7 and 7 has no square factor left:

The prime factorization tells the same story: 700 = 2² × 5² × 7. Each pair of equal primes leaves the radical as one factor, so 2 × 5 comes out and 7 stays inside.

700 has 3 square factors (4, 25 and 100). Starting with a smaller one still works but takes more rounds: √700 = 2√175, and √175 can be simplified again. Using 100 straight away finishes in one step.

Check: (10√7)² = 10² × 7 = 100 × 7 = 700. As a decimal, 10√7 = 10 × 2.6457513111 ≈ 26.4575131106.

Where √700 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √700 lies between 26 and 27. 700 is 24 above 676 and 29 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.4528 (0.02% low)
• Tangent from 26, i.e. 26 + 24 ÷ 52: 26.4615 (0.02% high)
• Tangent from 27, i.e. 27 − 29 ÷ 54: 26.4630 (0.02% high)

For √700 the tangent at 26 wins, missing by only 0.004. Tangent estimates shine when the number sits close to a perfect square — here 700 is just 24 above 676.

Finding √700 with the Babylonian method

Picture a rectangle with an area of 700 and one side x; the other side must be 700 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √700.

Start from the nearest whole number, 26 (26² = 676):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √700 = 26.4575131106 to every decimal shown.

√700 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √700 the pattern is [26; 2, 5, 2, 1, 1, 1, 1, 12, 1, 1, 1, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √700 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 − 700y² = 1. Its smallest solution in positive whole numbers is x = 8,193,151, y = 309,672.

√700 in geometry and everyday measurements

• 700 square feet is 65 m². Laid out as a square — a small house footprint or a lot — it is about 26.46 ft (26 ft 5 in) on a side.
• 700 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √700 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √700 as its space diagonal.
• Since √700 = 10√7, a length of √700 is exactly 10 copies of the length √7 laid end to end.

Square roots near √700 and related roots

• The cube root of 700 is about 8.879040.
• Dividing by 100 divides the root by 10: √7 = √700 ÷ 10 ≈ 2.64575131.

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