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Chapter 1: Problem 21

Evaluate each following expression for \(x=2, y=5,\) and \(z=3 .\) \(x^{5}+(y-z)\)

Short Answer

Expert verified
The expression evaluates to 34 for x=2, y=5, z=3.

Step by step solution

01

Substitute Values

First, substitute the given values for each variable in the expression. For this problem, substitute \(x = 2\), \(y = 5\), and \(z = 3\) into the expression \(x^5 + (y - z)\).
02

Evaluate Power of x

Next, calculate \(2^5\) since \(x = 2\). The expression \(2^5\) becomes \(2 \times 2 \times 2 \times 2 \times 2\), which results in 32.
03

Compute Difference y - z

Calculate \(y - z\) by subtracting \(z = 3\) from \(y = 5\). This results in the value \(5 - 3 = 2\).
04

Add the Results

Now, add the results from Step 2 and Step 3. This will be \(32 + 2\), which equals 34.

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Key Concepts

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

Substitution Method
The substitution method is a crucial concept in algebra that involves replacing variables with their given values to simplify expressions. When faced with an expression like \(x^5 + (y-z)\), the substitution method helps us to evaluate it by following these steps:
  • Identify the variables: First, note down the variables in the expression, which are \(x\), \(y\), and \(z\) in this case.
  • Select substitution values: The problem provides specific values for each variable: \(x = 2\), \(y = 5\), and \(z = 3\).
  • Replace and rewrite: Substitute these values into the expression鈥攖ransform \(x^5 + (y-z)\) into \(2^5 + (5-3)\).
By using this method efficiently, you simplify the task to just calculations with numbers instead of dealing with abstract variables.
Power Calculations
Power calculations in mathematics deal with the repeated multiplication of a number. It is denoted by a base number and an exponent. In our problem, this concept is illustrated by calculating \(2^5\), where \(2\) is the base and \(5\) is the exponent.
  • Understanding exponents: The exponent \(5\) tells us how many times to multiply the base \(2\) by itself.
  • Step by step calculation: Calculate \(2 \times 2 \times 2 \times 2 \times 2\). Start with two \(2's\): \(2 \times 2 = 4\). Multiply the result by \(2\) again: \(4 \times 2 = 8\). Keep this process until you perform all multiplications, reaching \(32\) for \(2^5\).
Understanding power calculations is key to solving problems involving exponents, as it allows you to handle complex expressions more efficiently.
Addition of Integers
Adding integers is a fundamental arithmetic operation that combines numbers to result in a sum. When evaluating algebraic expressions, after substituting variables and simplifying parts like differences, addition often comes into play.
  • Concept of Addition: When adding two numbers, such as \(32\) and \(2\), we combine them into a single number by establishing their total.
  • Calculation in the problem: In our example, after solving \(2^5\) to calculate \(32\), and finding \(y - z\) as \(2\), the final step involves summing these two results: \(32 + 2\).
Addition is straightforward but vital, especially when combining results from different operations like powers and subtractions in algebra.
Variable Evaluation
Variable evaluation refers to the process of determining the numeric outcome of an algebraic expression by substituting all variables with specified values. It combines the concepts discussed above.
  • Importance: This approach transforms variables, which can pose challenges, into concrete values, thereby simplifying mathematics.
  • Example execution: In the exercise \(x^5 + (y - z)\), evaluating each variable involves both substituting with \(x=2\), \(y=5\), and \(z=3\) and calculating each part鈥攑owers first, then differences, followed by summation.
  • Step-by-step simplification: First, apply power calculations to \(x^5\), then handle subtraction in \(y-z\), and finally, carry out the addition to find the final answer \(34\).
Variable evaluation is a comprehensive strategy that uses arithmetic to simplify algebraic expressions and obtain solutions easily.

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