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Simplifying algebraic expressions is a fundamental skill in algebra that involves reducing expressions to their simplest form. When an expression is simplified, it becomes easier to work with, as unnecessary complexities are removed. Let's break this down using the exercise as an example.

For expression (a) \(5(3x)\), we apply a straightforward multiplication, resulting in \(15x\). The process involves multiplying 5 by 3, and then the result is multiplied by \(x\).

Similarly, for (b) \(4(2x)\), multiplying the coefficients 4 and 2 gives us 8, and thus the expression simplifies to \(8x\). For (c) \(8(-7x)\), multiplying 8 by -7 results in -56, giving us the simplified expression \(-56x\).

By reducing these expressions, we transform complex terms into straightforward and manageable algebraic forms, streamlining calculations and problem-solving.

Multiplication in algebra involves multiplying numbers and variables to simplify expressions. It is an operation that combines values to form a new product. The exercise demonstrates multiplication between coefficients and the variable \(x\).

The distributive property plays a crucial role. It states that \(a(b + c) = ab + ac\), distributing the multiplication over addition. However, when no addition is present, as in \(a(b)\), it simplifies to straightforward multiplication: \(ab\).

For instance, multiplying 5 and 3 in \(5(3x)\) results in 15, making the expression \(15x\). Here, 5 multiplies the term \(3x\), simplifying to a singular term, \(15x\).

This approach remain the same for any variables or numbers, ensuring the algebraic expressions are simplified efficiently, leaving recognizably simpler forms.

Understanding step-by-step solutions enhances your ability to solve algebraic problems independently. It helps break down complex tasks into manageable steps. The exercise illustratively shows how even tricky problems are tackled logically.

Begin by using the distributive property to simplify expressions, as seen in all parts of the exercise.

• For (a), distribute 5 over \(3x\), leading to \(15x\).
• For (b), distribute 4 over \(2x\), giving \(8x\).
• For (c), 8 over \(-7x\) results in \(-56x\).

Each step resolves part of the problem. Keep your focus on multiplying coefficients and the variable until the expression is fully simplified. The process teaches a methodical way to approach various algebraic problems. This clarity in problem-solving builds confidence and proficiency in handling algebraic expressions.