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Understanding square roots is essential for solving many algebra problems. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9.

In notation, the square root of a number is represented as \( \sqrt{n} \). It is important to know that every positive number has two square roots: a positive and a negative (e.g., \( \sqrt{9} = \pm 3 \)). Generally, when we talk about square roots, we refer to the positive root unless otherwise stated.

Note: Square roots of non-perfect squares (like 7) result in irrational numbers with non-repeating decimal sequences. We often approximate such values for simplicity. For example, \( \sqrt{7} \approx 2.646 \).

• Always simplify inside the radical first before taking the square root.
• Use approximate values for quick comparisons but be aware of the exact irrational values.

Inequality comparisons involve determining the relationship between two expressions. The main symbols used are:

• \( < \) less than
• \( > \) greater than
• \( = \) equal to
• \( \leq \) less than or equal to
• \( \geq \) greater than or equal to

These symbols are used to compare the values on the left and right sides of the inequality. For example, \( 3 > 2 \) means 3 is greater than 2.

When working with square roots and other expressions, it is important to first simplify both sides of the inequality or equation before comparing them.

In example (a), we simplified each side separately: \( \sqrt{3} + \sqrt{7} \approx 4.378 \) and \( \sqrt{10} \approx 3.162 \). We then compared the simplified results: \( 4.378 > 3.162 \), leading to \( \sqrt{3} + \sqrt{7} > \sqrt{10} \).

• Always simplify and approximate where necessary before making comparisons.
• If both sides involve square roots, try approximating the values.
• Double-check calculations to ensure accuracy in comparisons.

Simplifying expressions makes it easier to solve equations and inequalities. Simplification involves breaking down complex expressions into their simplest forms.

Let's see how this applies:
In example (b), we started with \( \sqrt{3^{2} + 2^{2}} \). We simplified the inside of the square root: \( 3^{2} = 9 \) and \( 2^{2} = 4 \). Therefore, \( 3^{2} + 2^{2} = 9 + 4 = 13 \), resulting in \( \sqrt{13} \approx 3.606 \).

Similarly, in example (c), we dealt with \( \sqrt{5^{2} - 4^{2}} \). We simplified inside the square root: \( 5^{2} = 25 \) and \( 4^{2} = 16 \). Thus, \( 5^{2} - 4^{2} = 25 - 16 = 9 \), giving us \( \sqrt{9} = 3 \).

• Always perform operations inside parentheses or roots first.
• Combine like terms where possible before taking square roots.
• Simplify step-by-step to avoid errors and to make comparisons easier.

In summary, simplify expressions as much as possible to make analysis and comparison straightforward.