Square Root of 175

√175 at a glance

Exact value

5√7

Decimal (10 places)

13.2287565553

Rounded

13.2 · 13.23 · 13.229

Perfect square?

No — between 13² and 14²

Rational?

Irrational

Both square roots

±13.228757

Prime factorization

5² × 7

Cube root

5.593445

How to simplify √175

Look for the largest perfect square that divides 175. Here it is 25 (5²), because 175 = 25 × 7 and 7 has no square factor left:

The prime factorization tells the same story: 175 = 5² × 7. Each pair of equal primes leaves the radical as one factor, so 5 comes out and 7 stays inside.

Check: (5√7)² = 5² × 7 = 25 × 7 = 175. As a decimal, 5√7 = 5 × 2.6457513111 ≈ 13.2287565553.

Where √175 sits between perfect squares

169 = 13² and 196 = 14² are the nearest perfect squares, so √175 lies between 13 and 14. 175 is 6 above 169 and 21 below 196, so the root is closer to 13.

• Straight line between 169 and 196: 13.2222 (0.05% low)
• Tangent from 13, i.e. 13 + 6 ÷ 26: 13.2308 (0.02% high)
• Tangent from 14, i.e. 14 − 21 ÷ 28: 13.2500 (0.16% high)

For √175 the tangent at 13 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 175 is just 6 above 169.

Finding √175 with the Babylonian method

If a guess is too big, 175 ÷ 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√175) in one step.

Start from the nearest whole number, 13 (13² = 169):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √175 = 13.2287565553 to every decimal shown.

√175 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √175 the pattern is [13; 4, 2, 1, 2, 4, 26] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √175 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 − 175y² = 1. Its smallest solution in positive whole numbers is x = 2,024, y = 153.

√175 in geometry and everyday measurements

• A square room or garden bed covering 175 square feet measures about 13.23 ft (13 ft 3 in) along each wall.
• 175 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 √175 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 √175 as its space diagonal.
• Since √175 = 5√7, a length of √175 is exactly 5 copies of the length √7 laid end to end.

Square roots near √175 and related roots

• The cube root of 175 is about 5.593445.
• Four times the radicand doubles the root: √700 = 2 × √175 ≈ 26.457513.

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