Perfect Squares
- What Are Perfect Squares?
- Visualization of Perfect Squares
- Finding the Square Root of Perfect Squares
- Methods to Calculate Perfect Square
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What Are Perfect Squares?
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`.
Definition of Perfect Squares
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.
Visualization of Perfect Squares
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 the Square Root of Perfect Squares
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.
Methods to Calculate 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:
- Take tens digit i.e. `2` and multiply it by its next whole number i.e., `3` to get `6`.
- Take the digit at units place and square it i.e., `5^2=25`.
- Merge the numbers to get the perfect square i.e., `625`.
Solved Examples
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.
Practice Problems
Q`1`. Find two perfect squares that are between `60` and `100`.
- `49` and `64`
- `64` and `81`
- `81` and `98`
- `100` and `50`
Answer: b
Q`2`. Find the perfect square of number `103`.
- `9609`
- `8789`
- `10609`
- `6779`
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.
- `34` units
- `24` units
- `64` units
- `14` units
Answer: b
Frequently Asked Questions
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.