\(\underline{\mathrm{a}} \cdot(2 \underline{\mathrm{b}}+2 \mathrm{c}) \times(3 \underline{\mathrm{a}}+3 \underline{\mathrm{b}}+3 \underline{\mathrm{c}})=\) (a) \([\underline{\mathrm{a}} \underline{\mathrm{b}} \underline{\mathrm{c}}]\) (b) \(3[\underline{\mathrm{a}} \underline{\mathrm{b}} \underline{\mathrm{c}}]\) (c) \(6[\underline{\mathrm{a}} \underline{\mathrm{b}} \underline{\mathrm{c}}]\) (d) 0

Algebraic expressions are mathematical phrases involving numbers, variables, and operation symbols. They're like sentences in the language of mathematics that represent values that can change. The expression \(\underline{\mathrm{a}} \cdot(2 \underline{\mathrm{b}}+2 \mathrm{c}) \times(3\underline{\mathrm{a}}+3 \underline{\mathrm{b}}+3 \underline{\mathrm{c}})\) is a complex algebraic expression involving variables \(\underline{\mathrm{a}}\), \(\underline{\mathrm{b}}\), and \(\mathrm{c}\), as well as arithmetic operations of multiplication and addition.

Understanding these expressions is essential in algebra, as they form the basis for creating models to solve real-world problems. Simplifying algebraic expressions, which means reducing them to their simplest form, makes it easier to interpret or solve mathematical problems. When looking at an algebraic expression, breaking it down into its components and understanding the relationship between variables is the first step to mastery.

Preparing for competitive exams like the Joint Entrance Examination (JEE) in India challenges students to tackle a wide range of mathematical concepts, including proficiency in manipulating algebraic expressions. Problems in the JEE Maths section can be complex, requiring a deep understanding of various mathematical principles and their applications.

In the quest to achieve a high score, students must become adept with strategies like simplifying expressions and efficiently applying properties such as the distributive property. The expression given in the exercise is typical of the multistep algebraic problems encountered in JEE Maths. By practicing such problems regularly, students develop the essential skills required for the high-stakes environment of the JEE Maths section. Building a solid foundation in simplifying complex algebraic expressions is crucial for success on the exam and further mathematical studies.

The process of simplifying expressions can be thought of as a way to 'tidy up' the mathematical language. Taking a complicated expression and breaking it down into a more manageable form not only makes it easier to work with but also reveals its core components. To simplify expressions, one often utilizes properties of operations, such as the distributive property, which allows us to multiply a single term across terms within parentheses, as seen in our textbook exercise.

By decomposing the given expression through the distributive property, we effectively simplify it into a series of terms that can be easily manipulated or factored further. This methodical approach is vital, for instance, when solving equations, evaluating expressions, or finding polynomial roots. Practice is key, and with each problem solved, the principles of simplifying expressions become more intuitive, laying the foundation for solving even more complex algebraic equations.