Question

Evaluate the following expressions for \(x=2, y=-3,\) and \(z=-5\) $$ -y^{x} $$

Short answer

Question: Evaluate the expression \(-y^{x}\) when \(x=2, y=-3,\) and \(z=-5\). Answer: -9

Step-by-step solution

Write down the given values

Given \(x=2, y=-3,\) and \(z=-5\), the expression to evaluate is: \(-y^{x}\).

Substitute the given values into the expression

Now, substitute the values of \(x\) and \(y\) into the expression: $$-(-3)^{2}$$

Apply the exponent

Apply the exponent (2) to the base (-3): $$-(-3)^2 = -(9)$$

Calculate the final result

Now simply multiply the negative sign with the result from step 3: $$-(9) = -9$$ The evaluated expression for \(x=2, y=-3,\) and \(z=-5\) is: \(-y^{x} = -9\).

Key concepts

Substituting Values

Substituting values into an algebraic expression is a crucial step in evaluating expressions. It involves replacing variables with their given numerical values. For example, if an expression includes the variable \(y\), and you are provided that \(y = -3\), you substitute \(-3\) in place of \(y\). In this exercise, besides \(y\), we also have the variable \(x\), which is set to \(2\). Even though \(z = -5\) is given, this value is not used in the specific expression \(-y^{x}\). When substituting, ensure that each variable in the expression is replaced correctly:

• Replace \(y\) with \(-3\)
• Replace \(x\) with \(2\)

This practice allows you to convert the algebraic expression to a numerical expression that can be calculated through regular arithmetic operations.

Applying Exponents

After substituting values, the next step is to deal with exponents. An exponent refers to the number of times you multiply a number by itself. In our expression, this involves raising \(-3\) to the power of \(2\), written as \((-3)^2\). Here's how to apply exponents correctly:

• Identify the base, which is the number that is being multiplied by itself. Here it is \(-3\).
• Identify the exponent, which tells you how many times to multiply the base by itself. Here it is \(2\).

To compute \((-3)^2\), you multiply \(-3\) by itself:\[ (-3) imes (-3) = 9 \]It's important to note that a negative number raised to an even exponent results in a positive number. Therefore, \((-3)^2\) gives a positive \(9\).

Negative Numbers

Working with negative numbers can sometimes be tricky, especially when they interact with operations like exponents and multiplication. In this exercise, after applying the exponent, we had \(-(-3)^2\), which simplifies to \(-(9)\). The negative sign in front of \(9\) indicates that we need to multiply \(9\) by \(-1\).Important points about negative numbers:

• Multiplying a positive number by a negative number results in a negative number.
• The negative sign in front of an expression like \(-y^x\) affects the entire value of the power. Here, it means you take the negative of whatever \(y^x\) evaluates to.

Thus, multiplying \(9\) by \(-1\) results in \(-9\). This is the final step in evaluating our expression to get the correct result. Understanding how negative numbers interact with other arithmetic operations is vital in solving such problems accurately.