A square root of a number *b* is a solution of the equation *x*^{2} = *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 5^{2} = 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...)

**Example:**

- 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 = 3.271108This is close to the exact answer of 3.2710854...

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