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In mathematics, a perfect square is a number that is the square of an integer. For instance, when you multiply an integer by itself, you get a perfect square. This concept is vital when working with square roots because knowing which numbers are perfect squares can significantly simplify your calculations. For example:

• 81 is a perfect square because it equals 9 \( \times \) 9.
• 64 is a perfect square because it equals 8 \( \times \) 8.

In the exercise, we saw the square root of 81 and 64. Since these are perfect squares, their respective square roots simplify nicely:

• \( \sqrt{81} = 9 \)
• \( \sqrt{64} = 8 \)

Recognizing perfect squares allows you to convert complex square root expressions into more manageable multipliers. This process simplifies your work significantly and is a critical step in breaking down complex expressions.

Simplifying expressions involves reducing them to their most manageable form. This process is essential in algebra as it helps in solving equations and makes calculations easier. When simplifying square root expressions involving perfect squares and variables, follow these steps:

• Identify if numbers within the square root are perfect squares.
• Simplify the square root of those perfect squares first.
• Rewrite the expression by factoring out common terms.

As shown in the exercise, we encounter the square roots \( \sqrt{81ab} \) and \( \sqrt{64ab} \):

• First, simplify the coefficients: \( \sqrt{81ab} = 9\sqrt{ab} \) because \( 81 \) is a perfect square.
• Then, \( \sqrt{64ab} = 8\sqrt{ab} \) because \( 64 \) is a perfect square.

After simplification, combine terms that have common variables or components, such as \( 9\sqrt{ab} \) and \( 8\sqrt{ab} \), to arrive at a simpler expression. This practice is the cornerstone of working efficiently with complex mathematical problems.

Algebraic expressions are combinations of variables, numbers, and operations. They are the language of algebra and are used to represent relationships and operations mathematically.
Understanding how to work with these expressions is crucial. In algebra, many expressions can be simplified if you recognize patterns such as perfect squares and common factors.
In the given exercise, the expression involves terms like:

• \( 9\sqrt{ab} \)
• \( 8\sqrt{ab} \)

These are examples of algebraic expressions that include both coefficients and variables. By recognizing the common term \( \sqrt{ab} \), you can easily combine the expressions:

• \( 9\sqrt{ab} - 8\sqrt{ab} = (9-8)\sqrt{ab} \)

This results in a new, much simpler expression: \( \sqrt{ab} \).
Through combining like terms and simplifying expressions, you can solve algebraic problems more efficiently and understand the underlying mathematics more deeply.