A power of a number is generally expressed by writing the number only
once, and placing after it, above the line, a small figure to show how
many factors are to be taken. The small figure is called an _index_.
Thus, 7² = 49; 7³ = 343; 7⁴ = 2401.
A number is called the _square root_ of its square.
Since 7² = 49, the square root of 49 is 7.
The “square root of 49” is written √49.
Again, a number is called the _cube root_ of its cube. 7³ = 343.
Therefore, the cube root of 343 is 7.
The “cube root of 343” is written ∛343.
A _perfect square_ is a number whose square root is a whole number. A
_perfect cube_ is a number whose cube root is a whole number.
SQUARE ROOT.--If a number can be put into prime factors, its square root
can be written down by inspection.
EXAMPLE: Find the square root of 27225.
Since 27225 = 3² × 5² × 11².
∴ √27225 = 3 × 5 × 11 = 165 _Ans._
RULE FOR DIGITS.--We know that √1 = 1, and √100 = 10. Therefore, the
square root of any number which lies between 1 and 100 lies between 1
and 10; _i.e._, if a number contains one or two digits, its square root
consists of one digit.
Similarly, since √100 = 10 and √10000 = 100, the square root of a number
between 100 and 10000 lies between 10 and 100. That is, if a number
contains three or four digits, its square root consists of two digits.
Proceeding in this way, we obtain a general result--viz., the square of
a number has either twice as many digits as the number, or one less than
twice as many.
Hence, to ascertain the number of digits in the square root of a perfect
square, mark off the digits in pairs, beginning from the right. Each
pair marked off gives a digit in the square root; and, if there is an
odd digit remaining, that digit also gives a digit in the square root.
EXAMPLES: There are three digits in the square root of 546121, and four
in the square root of 5774409.
For, marking off the digits from the right, we get in the first case
54,61,21, giving three digits in the square root, and in the second case
5,77,44,09, the odd digit giving the fourth in the square root.
The method of finding the square root of a given number depends on the
_form_ of the square of the sum of two numbers.
EXPLANATION: The square root of 144 is 12. Let us see how we found it.