Perfect squares are obtained when whole numbers or integers are multiplied by itself. All perfect squares are positive integers. The perfect square of any integer `x` is written as `x^2`.
Here `2` is the exponent of the integer `x` whose square is being calculated.
For example, `4` is the perfect square of `2` and it can be written as `2^2=2\times 2=4`.
Also, `4` is also the perfect square of `(-2)`. It can be seen as `(-2)\times (-2)=4`.
A perfect square is defined as a non-negative number that is obtained by multiplying integers (either positive or negative) by itself.
The following table represents the perfect squares of integers `1` to `10`. We often use perfect square numbers while studying radicals, exponents, logarithms, etc.
Perfect squares are very important in math as they are found in applications of areas, volumes, number theory, etc.
For visualization of perfect squares consider small squares of unit side lengths as shown in the image below. Note that `9` small unit squares can be arranged in a big square with a side length of `3` units.
The area of the big square can be calculated as `\text{side length}^2=3^2=9` `\text{unit}^2`. Here, `9` `\text{unit}^2` denotes that the big square can accommodate `9` squares of unit area.
This can be replicated for squares with side lengths of `4` units, `5` units, `6` units, etc,.
Other examples can be tiles of a floor, chessboards, brick walls, etc.
Finding square root of a number is an easy way to identify whether the given number is a perfect square or not. Square root of a number is a value that we can multiply by itself to get the original number. A square root is denoted by a radical sign `\sqrt{\quad}`.
Example `1`: Identify whether `84` is a perfect square or not.
Solution:
`\sqrt{84}=+9.165` and `-9.165`
The square root of `84` are not integers. Therefore, `84` is not a perfect square.
Example `2`: Identify whether `324` is a perfect square or not.
Solution:
`\sqrt{324}=+18` and `-18`
The square root of `324` are integers. Therefore, `324` is a perfect square.
`1`. Finding squares by the formula.
The formula that is used to calculate the perfect square is as follows:
`(a+b)^2=a^2+2ab+b^2`
This formula can be used to find the squares of large numbers.
For example, the square of `76` can be calculated as
\(\begin{align*}
(76)^2 & = (70 + 6)^2 \\
& = 70^2 + 2(70)(6) + 6^2 \\
& = 4900 + 840 + 36 \\
& = 5776
\end{align*}\)
`2`. Calculating perfect squares of two-digit numbers ending with `5`.
Suppose, we have to calculate square of `25`.
The above image can be summarized as follows:
Example `1`: Find the square of `83`.
Solution:
We can use the formula `(a+b)^2=a^2+2ab+b^2` to calculate the square.
\(\begin{align*}
(83)^2 & = (80 + 3)^2 \\
& = 80^2 + 2(80)(3) + 3^2 \\
& = 6400 + 480 + 9 \\
& = 6889
\end{align*}\)
Example `2`: Find the perfect square of `65`.
Solution:
Since it is a number ending with `5`. We can calculate the square by the trick as shown in the image above.
Q`1`. Find two perfect squares that are between `60` and `100`.
Answer: b
Q`2`. Find the perfect square of number `103`.
Answer: c
Q`3`. A square garden was measured in terms of area. If the area of the garden is measured to be `576` `\text{unit}^2`. Find the dimension of the square field.
Answer: b
Q`1`. Do negative numbers have perfect squares?
Answer: Yes, negative numbers have perfect squares. The square of a negative number is the same as the square of its positive counterpart. This is because negative times a negative also gives a positive result. For example, the perfect square of `-12` is `144`.
Q`2`. Are there any practical applications involving perfect squares?
Answer: Yes, there are numerous applications involving perfect squares such as in geometry, physics, engineering, Pythagoras' theorem in mathematics, number theory, medicine, digital storage, etc.