Square Root of 34: √34 = 5.8310 (Long Division Method)

Simplest radical form

34 has no square factor other than 1, so √34 is already in its simplest form.

Group the digits in pairs from the decimal point: 34. The largest square ≤ 34 is 5² = 25, so the first digit is 5; remainder 9.
Bring down 00: 900. Double the root so far (5 → 10) and find the largest d with (10d) × d ≤ 900: d = 8, since 108 × 8 = 864. Remainder 36.
Bring down 00: 3600. Double the root so far (58 → 116) and find the largest d with (116d) × d ≤ 3600: d = 3, since 1163 × 3 = 3489. Remainder 111.
Bring down 00: 11100. Double the root so far (583 → 1166) and find the largest d with (1166d) × d ≤ 11100: d = 0, since 11660 × 0 = 0. Remainder 11100.

The digits read 5.830, so √34 ≈ 5.830.

5² = 25 and 6² = 36, so √34 lies between 5 and 6. A fast first guess is 5 + 9/(2 × 5) = 5.900, against the true 5.831.

What is the square root of 34?

√34 ≈ 5.8310. It is irrational, so the decimal never ends or repeats.

Is √34 rational?

No. 34 is not a perfect square, so √34 is irrational.

What is √34 in simplest radical form?

34 has no square factor other than 1, so √34 is already in its simplest form.

How do you find √34 by long division?

Pair the digits, find the largest square under the first pair, then repeatedly double the root so far and find the next digit. It gives √34 ≈ 5.830.