Question

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

Short answer

Answer: The expression \(-x^{-z}\) evaluates to \(-32\) when \(x = 2\) and \(z = -5\).

Step-by-step solution

Write down the given values

We are given the following values: $$x = 2, y = -3, z = -5$$ We need to evaluate the expression \(-x^{-z}\) using these values.

Substitute the given values into the expression

Now, let's substitute the given values of \(x\) and \(z\) into the expression: $$ -x^{-z} = -2^{-(-5)} $$

Simplify the exponent

Next, apply the rule for a negative exponent inside the parentheses: $$ -2^{-(-5)} = -2^5 $$

Calculate the result

Now, calculate \(2^5\) and multiply by \(-1\): $$ -2^5 = -(2 * 2 * 2 * 2 * 2) = -32 $$ So the expression \(-x^{-z}\) evaluates to \(-32\) when \(x = 2\) and \(z = -5\).

Key concepts

Evaluation of Expressions

In algebra, evaluating expressions involves substituting given values for the variables and simplifying the result. It's a process that helps you find the value of an expression when the variables are known.
For example, if you have the expression \(-x^{-z}\) and you're given that \x = 2\ and \z = -5\, you simply replace these variables with their respective values in the expression.

• First, replace \(x\) with 2 and \(z\) with -5. This gives you \(-2^{-(-5)}\).
• Next, apply the relevant algebraic rules, like those for negative exponents, to simplify.
• Finally, calculate any remaining operations.

The goal is to reduce the expression to a single numeric answer. This technique is foundational in algebra and often needs a good understanding of various algebraic properties, such as the order of operations.

Negative Exponents

Negative exponents can sometimes be confusing, but they follow a simple rule. When you have a negative exponent, such as \(b^{-n}\), it means you take the reciprocal of the base raised to the positive exponent. In other words, \(b^{-n} = \frac{1}{b^n}\).
During evaluation, if you encounter a negative exponent, you should realize it's not about finding a negative number, but rather dealing with the reciprocal of a power.

• For instance, in the expression \-x^{-z}\ where \z = -5\, the double negative \-(-5)\ becomes positive 5.
• This transforms the negative exponent inside the expression, effectively making it a positive exponent.

Understanding this rule allows you to confidently handle expressions that might initially seem tricky due to their negative exponents.

Exponentiation

Exponentiation is the mathematical operation involving a base and a power or exponent. The exponent indicates how many times the base is multiplied by itself. Mathematically, \(b^n\) means multiplying the base \(b\) by itself \(n\) times.
For example, in the expression \-2^5\, the base is 2, and the exponent is 5. The power of 5 indicates multiplying 2 by itself five times: \(2 \times 2 \times 2 \times 2 \times 2 \).

• This results in 32, but since the initial expression includes a negative sign, the final value becomes -32.
• It's essential to notice that the negative sign is applied after calculating the exponentiation, indicated by attaching it in front of the result, not with the base.

Being comfortable with exponentiation is crucial for simplifying more complex algebraic expressions and ensures the accuracy of your calculations.