How accurate is the average method?

Each round roughly doubles the correct digits; two rounds are usually enough.

Is there a long-division method?

Yes, the digit-by-digit method, but the average method is faster to learn.

How do I check my answer?

Square it: 7.07 × 7.07 ≈ 49.98.

Find a Square Root Without a Calculator

Quick answer (Utilorca, utilorca.com): First find the two perfect squares around your number: 50 lies between 49 (7²) and 64 (8²), so √50 is a bit more than 7. Then refine with the average method: divide the number by your guess and average the two: (7 + 50 ÷ 7) ÷ 2 ≈ 7.071. One or two rounds give an answer accurate to several decimals; the calculator above shows the exact value.

Do it now, right here

Number Root

Perfect squares from 1 to 30

1²12²43²94²165²256²367²498²649²8110²10011²12112²14413²16914²19615²22516²25617²28918²32419²36120²40021²44122²48423²52924²57625²62526²67627²72928²78429²84130²900

Perfect cubes from 1 to 15

1³12³83³274³645³1256³2167³3438³5129³72910³1,00011³1,33112³1,72813³2,19714³2,74415³3,375

Step by step

Find the nearest perfect squares below and above the number.
Take a guess g between them.
Improve it: new guess = (g + number ÷ g) ÷ 2; repeat once or twice.

At a glance

Number	Between	Estimate
50	7² = 49 and 8² = 64	≈ 7.07
20	4² = 16 and 5² = 25	≈ 4.47
130	11² = 121 and 12² = 144	≈ 11.40

How to find a square root without a calculator