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Chapter 2: Problem 109

Evaluate. $$ (x y+z)^{x} \text { for } x=2, y=-5, \text { and } z=7 $$

Short Answer

Expert verified
The result is 9.

Step by step solution

01

Write the Expression

We are given the expression \((xy + z)^x\) and need to evaluate it for \(x = 2\), \(y = -5\), and \(z = 7\).
02

Substitute the Values

Replace \(x\), \(y\), and \(z\) in the expression with the given values. The expression becomes \(((2)(-5) + 7)^2\).
03

Simplify the Inside of the Parentheses

Calculate the value inside the parentheses first: \((2)(-5) = -10\) and then add \(7\) to \(-10\) to get \(-3\).
04

Compute the Power

The expression now is \((-3)^2\). This means we multiply \(-3\) by itself: \((-3) \times (-3) = 9\).
05

Write the Final Result

The evaluated result of the expression \((xy + z)^x\) for the given values is \(9\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Expressions
An algebraic expression is like a mathematical sentence that uses variables, numbers, and operations. You can think of it as a combination of numbers and letters, where the letters represent variables, or unknown values. For example, in the expression \((xy + z)^x\), the letters \(x\), \(y\), and \(z\) are variables. Algebraic expressions are central to algebra and allow us to write equations and functions in a general form. This generality is powerful, as it means we can solve problems for many different values just by substituting specific numbers in place of the variables. This leads us directly to the next important step: substitution.
Substitution
Substitution is the process where we replace the variables in an algebraic expression with specific values. This step is crucial for evaluating expressions. In our problem, we were given specific values for \(x = 2\), \(y = -5\), and \(z = 7\). Let's see how to do this:
  • First, identify the variables in the expression \((xy + z)^x\).
  • Next, replace each variable with its given value. For example, replace \(x\) with \(2\), \(y\) with \(-5\), and \(z\) with \(7\).
  • The expression becomes \(((2)(-5) + 7)^2\).
This process simplifies the expression into a form you can easily calculate. Substitution allows you to plug in numbers and make complex expressions manageable.
Order of Operations
Order of operations is a set of rules mathematics follows to evaluate expressions in the correct order. It's often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Applying the order of operations to our expression \(((2)(-5) + 7)^2\), we:
  • First, calculate anything inside parentheses: \((2)(-5) = -10\), then \(-10 + 7 = -3\).
  • Next, tackle the exponent: \((-3)^2\) which means multiply \(-3\) by itself: \(-3 \times -3 = 9\).
By following the order of operations strictly, you ensure accurate results. Remembering PEMDAS helps avoid common pitfalls in calculations.

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Most popular questions from this chapter

Graph each integer in the list on the same number line. $$ 0,1,9,14 $$

Find the opposite of each integer. $$ 123 $$

Find the opposite of each integer. $$ -6 $$

Insert \(<\) or \(>\) between each pair of integers to make a true statement. $$ -12 \quad-10 $$

Add. $$ 20+15 $$
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