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Understanding the order of operations is crucial when evaluating mathematical expressions. It ensures accuracy by providing a standard procedure for calculations. The order, often remembered by the acronym PEMDAS, stands for Parentheses, Exponents, Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right).

For instance, in the given problem \( (x+y)^{2} \) when \( x=5 \) and \( y=2 \) from the textbook, identifying the correct order is fundamental. The parenthesis \( (x+y) \) indicates that we must add the values of \( x \) and \( y \) before applying the exponent. Only after this initial addition can the exponent be computed, reflective of the 'E' in PEMDAS. By following these rules, errors can be significantly reduced, leading to a correct solution of 49 after applying the exponent to 7.

The substitution method is a powerful technique used in algebra to evaluate expressions or solve equations involving one or more variables. This method involves replacing variables with their known or assumed values to simplify or solve an equation.

In the context of our example, \( (x+y)^{2} \), we apply the substitution method by taking the given values for \( x \) and \( y \) (\( x=5 \) and \( y=2 \) respectively), and substituting them directly into the expression. This converts \( (x+y)^{2} \) into \( (5+2)^{2} \). Such a substitution transforms an expression with variables into a numerical expression, which can then be easily evaluated using arithmetic operations. This step is foundational and critical for solving more complex algebraic problems, demonstrating the importance of substitution in understanding the relationship between variables.

Exponent calculation is an operation involving a base number raised to the power of an exponent, indicating how many times the base is multiplied by itself. The expression \( a^{n} \) signifies the base \( a \) is used as a factor \( n \) times.

For example, in our textbook problem where we ended up with \( 7^{2} \) after substitution and addition, computing the exponent means multiplying 7 by itself: \( 7 \times 7 = 49 \). It's also important to remember that when an exponent is a positive integer, the calculation is straightforward. However, exponents can also be negative, zero, or even fractions, each presenting its own set of rules for calculation. Mastery of exponent rules enables students to accurately and efficiently handle more complex expressions in algebra and higher-level math courses.