Square Roots

How do we do them?

A square root of a number b is a solution of the equation x2 = b. Every number except 0 has two square roots, a positive and a negative.  The positive square root is the principal square root and is written .  To denote the negative root, write and to indicate both roots write .

So, we call 5 the “square root” of 25 and write because 52 = 25. (See exponents for more on this.)  Since (–5)2 also equals 25 it is also a “square root” of 25, but we write because it is not the principal square root.

Not all whole numbers have a whole number square root.  For instance (The decimal goes on forever and never repeats a pattern.  This is called an irrational number.)

How can you estimate the value of a square root like ? Well, you could first notice that and , since 64 and 81 are both perfect squares. 70 is in-between 64 and 81, so . Since 70 is closer to 64 than to 81, is closer to 8.

To find a square root more precisely without a calculator, you can use Newton's Divide and Average method:

1) Guess the square root of the number.

2) Divide the guess into the original number.

3) Take the Average of the guess and quotient.

4) Repeat with this average as new guess.

"Guess, Divide, Average, Repeat."(I guess GDAR isn't a great way to remember that...)


  • Guess: is about 3
  • Divide 3 into 10.7 and get 10.7 / 3 = 3.566666
  • Average (3 + 3.566666) / 2 = 3.283333
  • Repeat:
    10.7 / 3.283333 = 3.258884
    (3.283333 + 3.258884) / 2 = 3.271108

    This is close to the exact answer of 3.2710854...

But the way most people calculate square roots, these days, is to use a calculator.