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The substitution method is crucial in evaluating algebraic expressions when specific values are provided for the variables. When you substitute, you're essentially replacing a variable with its given numeric value. This simplifies the expression, making it manageable and ready for further operations.

The process typically involves:

• Identifying the variables in the expression
• Replacing each variable with its given value
• Rewriting the expression with these values in place of the variables
• Proceeding to calculate the simplified expression

In the given problem, we have the expression \(y^x\) and the values \(x = 2\) and \(y = -1\). By substituting \(x = 2\) and \(y = -1\), the expression becomes \((-1)^2\).

Understanding exponents is essential when simplifying expressions like the one in the problem. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \(a^b\), \(a\) is the base, and \(b\) is the exponent, meaning \(a\) is multiplied by itself \(b\) times.

Key points to remember about exponents:

• Any number raised to the power of 1 is itself: \(a^1 = a\)
• Any number (except zero) raised to the power of 0 is 1: \(a^0 = 1\)
• A negative base raised to an even exponent results in a positive value
• A negative base raised to an odd exponent results in a negative value

For the problem at hand, we simplify \((-1)^2\):
Since 2 is an even number, \((-1)^2\) equals 1, as shown by \((-1) * (-1) = 1\). Hence, the expression simplifies to 1.

Performing operations on integers is fundamental to solving many algebraic problems. Integers include positive and negative whole numbers, as well as zero. Basic operations involve:

• Addition
• Subtraction
• Multiplication
• Division

When multiplying and dividing integers, pay close attention to their signs. Multiplying or dividing two integers with the same sign (both positive or both negative) results in a positive product or quotient. Conversely, multiplying or dividing two integers with different signs results in a negative product or quotient.

For our example, we deal with the multiplication of \(-1\) by itself, which appears in the expression \((-1)^2\). Here, the negative integer \(-1\) is multiplied by another \(-1\), and since the signs are the same, the result is positive. Hence, \((-1) * (-1) = 1\). This fundamental understanding of integer operations ensures you correctly interpret and solve such expressions.