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Understanding the order of operations is crucial when solving algebraic expressions, as it outlines the sequence in which multiple arithmetic operations must be performed. The acronym PEMDAS is often used to help remember this sequence. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In our example, the expression \(4(5)^{2}\) requires us to address the exponent before we tackle the multiplication. This is in line with the 'E' for Exponents in PEMDAS. No matter how complex an expression is, always follow the order of operations to get the correct result. This systematic approach prevents errors and ensures consistency in solving mathematical problems.

Multiplication in algebra might seem straightforward, but it's important to understand its properties and how it interacts with other operations. When multiplying numbers and variables, remember that the operation of multiplication is commutative, meaning that the order in which you multiply the numbers does not change the product, and it is also associative, which allows you to group numbers in different ways without changing the result.

For instance, when we multiply \(4\) by \(25\) in the step-by-step solution above, we are using the commutative property. It doesn't matter if we write it as \(4 \times 25\) or \(25 \times 4\); the product will be \(100\). In algebra, always pay attention to coefficients (numerical factors) and how they interact with variables during multiplication.

Solving algebraic expressions often involves a combination of operations including exponentiation, multiplication, and more. To solve such expressions correctly, a deep understanding of the order of operations is necessary. Always look for and simplify expressions within parentheses first, evaluate exponents second, then proceed with multiplication and division, and finally tackle addition and subtraction.

When solving the given expression \(4(5)^{2}\), the exponent comes first because it is the next operation after considering the parentheses. By raising \(5\) to the power of \(2\) and then multiplying by \(4\), we simplify the expression step by step, following the right order to arrive at the correct answer, which in this case is \(100\). This methodical approach allows for clear and concise solving of algebraic terms, ultimately leading to the accurate simplification of expressions.