Question

Find the value of each expression if \(x=2\) and \(y=-1\) \(y^{x}\)

Short answer

The value of the expression is 1.

Step-by-step solution

- Understand the Problem

We need to find the value of the expression when specific values are given for the variables. Specifically, we need to find the value of the expression when \(x = 2\) and \(y = -1\) for the expression \(y^x\).

- Substitute the Values

Substitute \(x = 2\) and \(y = -1\) into the expression \(y^x\). This gives us \((-1)^2\).

- Calculate the Expression

Calculate \((-1)^2\). Since any number to the power of 2 is that number multiplied by itself, \((-1) * (-1) = 1\).

Key concepts

Substitution Method

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 \((-1)^2\).

Exponents

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.

Integer Operations

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.