Square Roots

How do we do them?

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 b .  To denote the negative root, write b and to indicate both roots write ± b .

So, we call 5 the “square root” of 25 and write 25 =5 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 25 =5 because it is not the principal square root.

Not all whole numbers have a whole number square root.  For instance 2 =1.414213562 (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 70 ? Well, you could first notice that 64 =8 and 81 =9 , since 64 and 81 are both perfect squares. 70 is in-between 64 and 81 , so 8< 70 <9 . Since 70 is closer to 64 than to 81 , 70 is closer to 8 .

To find a better approximation, you can use a calculator:

70 8.3666