Q&A for How to Find a Square Root Without a Calculator

There are two ways to do it: (1) If you can measure the inside diagonals (from a corner to its opposite corner), the diagonals of a perfect rectangle are equal to each other; (2) Get a magnetic compass and sight it along two adjacent sides. The two directions should be exactly 90° from each other.

Use a factor tree. For example, 625 = 5 x 125 = 5 x 5 x 25 = 5 x 5 x 5 x 5. Because there are 4 fives, and we are looking for the square root, (5 x 5)(5 x 5) = 625. Therefore the square root of 625 is 25.

The number 1089 is the answer. The way you work it out is 1008 is a 9 time table number. Then just add on nines starting at this number, and the first 9 times table number you get that has two even digits and odd digits is your answer.

Use the Pythagorean theorem to find the third side, then add the sides together.

To rationalize a denominator, multiply the fraction by a "clever" form of 1. In other words, you need a fraction whose numerator and denominator are both equal to the square root in the denominator.

There's only one step: multiply the number by itself. x² = (x)(x).

Since the square root of 24 is not an integer, you'll have to accept a decimal.

See Calculate a Square Root by Hand .

For cubed roots, the cubed root multiplied by itself 3 times will give you the number. So, mathematically, if the quotient obtained on dividing a number by any number is the square of the second number, that's your cubed root. For example, 216/6=36, which is the square of 6. Thus, 6 is the cube root of 216.

There are other ways, such as by prime factorization, repeated subtraction, long division or estimation, but Guess and Check is commonly used because it's probably the quickest and easiest method (once you get used to doing it).

Assuming you can't (or don't want to) use a calculator, we know that the square of 6 is 36, and the square of 7 is 49. So the square root of 44 will be roughly halfway between 6 and 7. So let's square 6.5: that's 42,25. So the number you're looking for is a little larger than 6.5. The square of 6.6 is 43.56. So we're almost there. The square of 6.65 is 44.22. So let's try squaring 6.64: that's 44.09. If that's not close enough for you, try squaring 6.635: that's 44.02. Now square 6.634: that's 44.01. The square of 6.633 is 43.997. So the square root of 44 is slightly larger than 6.633. (In fact, it's 6.6332.)

The square root of 2 is 1.414. 10 times that is 14.14.

40.

Not in any practical sense. You could use division in a process of trial-and-error, but it could prove to be quite cumbersome in many cases. (In a case where the number happened to be a perfect square, the trial-and-error process could be relatively quick and easy.

If you're not allowed to multiply (or use a calculator), you would have to add up 21 21's by hand.

Not that anyone's been able to discover.

That means that n is the square root of 9, which is 3.

Assuming you're not using a calculator, you could square 200 (that's 40,000), and to that add the square of 8 (which is 64). 40,000 + 64 = 40,064.

Unfortunately, no. The technique shown above is one way to do it without using a calculator, logarithm tables or a slide rule. There are other techniques, too, that consist of a series of successively closer estimations.

There's no shortcut, assuming you're not using a log table. Manually multiply the number by itself.