91Ó°ÊÓ

In mathematics, the square root of a number is a value that, when multiplied by itself, gives the original number. The square root is usually represented by the radical symbol (\(\sqrt{ }\)). For example, the square root of 9 is 3, because \(3 \times 3 = 9\). Similarly, the square root of 16 is 4, because \(4 \times 4 = 16\). Now, let’s consider the square root of 10. We need to find a number that, when squared, equals 10. Since \(3^2 = 9\) and \(4^2 = 16\), \The number 10 lies between 9 and 16. Therefore, \(\sqrt{10}\) must fall between 3 and 4.

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols like \(>\) (greater than), \(<\) (less than), \(\geq\) (greater than or equal to), and \(\leq\) (less than or equal to). For example, in the inequality \(9 < 10 < 16\), we see that 10 is greater than 9 and less than 16. This tells us that 10 is somewhere between these two values on the number line. By squaring the boundaries (3 and 4) and comparing the results (9 and 16) with 10, we use inequalities to find that \(3 < \sqrt{10} < 4\).
An important property of inequalities is that if you perform the same operation on both sides of an inequality, the relationship remains true.

A number line is a visual representation of numbers on a straight line, where each point corresponds to a number. It helps us understand and compare the size of numbers. Imagine a number line where 0 is at the center, positive numbers extend to the right, and negative numbers extend to the left. Now, let's focus on the segment between 3 and 4.
If we mark 9 at \(3^2\) and 16 at \(4^2\), 10 falls between these two points. This means that \(\sqrt{10}\), which represents the square root of 10, must be between 3 and 4 on the number line. Visualizing this on a number line helps in understanding that \(\sqrt{10} < 4 \) and \( \sqrt{10} > 3 \).

• This makes it easier to grasp why \(\sqrt{10}\) resides between 3 and 4.