Complete step-by-step answer:
$1057$
The number $1057$ends with 7 and does not end with either $1,4,5,6,9,00$ or even a number of 0’s.
Therefore, it cannot be a perfect square.
$23453$
The number $23453$ ends with 3 and does not end with either $1,4,5,6,9,00$ or even number of 0’s.
Therefore, it cannot be a perfect square.
$7928$
The number $7928$ ends with 8 and does not end with either $1,4,5,6,9,00$ or even a number of 0’s.
Therefore, it cannot be a perfect square.
$222222$
The number $222222$ ends with 2 and does not with either $1,4,5,6,9,00$ or even number of 0’s at the end.
Therefore, it cannot be a perfect square.
$64000$
The number $64000$ ends with three 0’s and does not with either $1,4,5,6,9,00$ or even number of 0’s at the end.
Therefore, it is not a perfect square.
The number $89722$ ends with 2 and does not with $1,4,5,6,9,00$ or even number of 0’s at the end.
Therefore, it is not a perfect square.
$222000$
The number $222000$ ends with 3 zero’s and not with $1,4,5,6,9,00$ or even number of 0’s.
Therefore, it is not a perfect square.
$505050$
The number $505050$ ends with only 1 zero and not with $1,4,5,6,9,00$ or even number of 0’s.
Therefore, it is not a perfect square.
Note: The number to be a perfect square should satisfy the first condition that it should end with either $1,4,5,6,9,00$ or even a number of zeroes.
Also, the digital roots of the number should be $1,4,7$or $9$ . Digital roots can be obtained by adding all the digits of a number and then again adding all its digits until it becomes a one digit number.
(i) 1296
We know that
It can be written as
1296 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3
Here
After pairing the same prime factors, no factor is left.
Therefore, 1296 is a perfect square of 2 × 2 × 3 × 3 = 36.
(ii) 1764
We know that
It can be written as
1764 = 2 × 2 × 3 × 3 × 7 × 7
Here
After pairing the same factors, no factor is left.
Therefore, 1764 is a perfect square of 2 × 3 × 7 = 42.
(iii) 3025
We know that
It can be written as
3025 = 5 × 5 × 11 × 11
Here
After pairing the same prime factors, no factor is left.
Therefore, 3025 is a perfect square of 5 × 11 = 55.
(iv) 3969
We know that
It can be written as
3969 = 3 × 3 × 3 × 3 × 7 × 7
Here
After pairing the same prime factors, no factor is left.
Therefore, 3969 is a perfect square of 3 × 3 × 7 = 63.
(i) 729
We know that
It can be written as
729 = 3 × 3 × 3 × 3 × 3 × 3
Here
729 is the product of pairs of equal prime factors
Therefore, 729 is a perfect square.
(ii) 5488
We know that
It can be written as
5488 = 2 × 2 × 2 × 2 × 7 × 7 × 7
Here
After pairing the same prime factors, one factor 7 is left unpaired.
Therefore, 5488 is not a perfect square.
(iii) 1024
We know that
It can be written as
1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
Here
After pairing the same prime factors, there is no factor left.
Therefore, 1024 is a perfect square.
(iv) 243
We know that
It can be written as
243 = 3 × 3 × 3 × 3 × 3
Here
After pairing the same prime factors, factor 3 is left unpaired.
Therefore, 243 is not a perfect square.
Formally, a string is a finite, ordered sequence of characters such as letters, digits or spaces. The empty string is the special case where the sequence has length zero, so there are no symbols in the string. There is only one empty string, because two strings are only different if they have different lengths or a different sequence of symbols. In formal treatments, the empty string is denoted with ε or sometimes Λ or λ.
The empty string should not be confused with the empty language ∅, which is a formal language (i.e. a set of strings) that contains no strings, not even the empty string.
The empty string has several properties:
In context-free grammars, a production rule that allows a symbol to produce the empty string is known as an ε-production, and the symbol is said to be "nullable".
Use in programming languages[edit]
In most programming languages, strings are a data type. Strings are typically stored at distinct memory addresses (locations). Thus, the same string (for example, the empty string) may be stored in two or more places in memory.
In this way, there could be multiple empty strings in memory, in contrast with the formal theory definition, for which there is only one possible empty string. However, a string comparison function would indicate that all of these empty strings are equal to each other.
Even a string of length zero can require memory to store it, depending on the format being used. In most programming languages, the empty string is distinct from a null reference (or null pointer) because a null reference points to no string at all, not even the empty string. The empty string is a legitimate string, upon which most string operations should work. Some languages treat some or all of the following in similar ways: empty strings, null references, the integer 0, the floating point number 0, the Boolean value false, the ASCII character NUL, or other such values.
The empty string is usually represented similarly to other strings. In implementations with string terminating character (null-terminated strings or plain text lines), the empty string is indicated by the immediate use of this terminating character.
Examples of empty strings[edit]
The empty string is a syntactically valid representation of zero in positional notation (in any base), which does not contain leading zeros. Since the empty string does not have a standard visual representation outside of formal language theory, the number zero is traditionally represented by a single decimal digit 0 instead.
Zero-filled memory area, interpreted as a null-terminated string, is an empty string.
Empty lines of text show the empty string. This can occur from two consecutive EOLs, as often occur in text files, and this is sometimes used in text processing to separate paragraphs, e.g. in MediaWiki.