Eight Standard >> Squares and squares root | Introduction

Square of a number

The term "square of a number" refers to the result of multiplying a number by itself. It is the number obtained when a number is raised to the power of 2. For any real number 'x', the square of 'x' is represented as \(x^2\)'.

For example:

The square of 5 is \(5^2\) = 5 * 5 = 25.
The square of -3 is \((-3)^2\) = (-3) * (-3) = 9.
The square of 1.5 is \((1.5)^2\) = 1.5 * 1.5 = 2.25.

A "perfect square" is a whole number that can be expressed as the square of another whole number.  A perfect square is the result of squaring an integer. Perfect squares are always non-negative because the square of any real number, positive or negative, is positive.

For example:

1 is a perfect square because 1 = \(1^2\).
4 is a perfect square because 4 = \(2^2\).
9 is a perfect square because 9 = \(3^2\).
16 is a perfect square because 16 = \(4^2\).
On the other hand, numbers like 2, 6, 10, and 15 are not perfect squares because they cannot be expressed as the square of any integer.

The "perfect square of a rational number" is the square of a rational number that results in a perfect square. It means that the square of the rational number will be a whole number, and therefore, it can be expressed as the square of an integer.

For example:

The perfect square of the rational number \(\frac{1}{2}\) is \((\frac{1}{2})^2\) = \((\frac{1}{2}) * (\frac{1}{2})\) = \(\frac{1}{4}\), which is not a perfect square because it is not a whole number.
The perfect square of the rational number \(\frac{3}{5}\) is \((\frac{3}{5})^2\) = \((\frac{3}{5}) * (\frac{3}{5})\) = \(\frac{9}{25}\), which is also not a perfect square because it is not a whole number.
However, the perfect square of the rational number \(\frac{2}{3}\) is \((\frac{2}{3})^2\) = \((\frac{2}{3}) * (\frac{2}{3})\) = \(\frac{4}{9}\), which is a perfect square because it is equal to \(\Big(\frac{2}{3}\Big)^2 = \frac{(2^2)}{(3^2)} = \frac{4}{9}\).

In summary, the square of a number is obtained by multiplying the number by itself. A perfect square is a whole number that can be expressed as the square of an integer. The perfect square of a rational number is the square of a rational number that results in a whole number.