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Chapter 1: Problem 15

Evaluate each following expression for \(x=2, y=5,\) and \(z=3 .\) \(y^{3}-4 x\)

Short Answer

Expert verified
The expression evaluates to 117.

Step by step solution

01

Substitute the Given Values

First, substitute the values of \(x\), \(y\), and \(z\) into the expression. We have \(x = 2\), \(y = 5\), and \(z = 3\). The expression is \(y^3 - 4x\), so substitute these values to get \(5^3 - 4 \times 2\).
02

Calculate the Exponent

Next, calculate \(5^3\), which means \(5 \times 5 \times 5\). This results in 125.
03

Evaluate the Multiplication

Now, compute \(4 \times 2\), which is equal to 8.
04

Solve the Expression

With the calculations done, substitute back into the expression: \(125 - 8\).
05

Final Calculation

Subtract 8 from 125 to get 117.

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Key Concepts

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

Exponents and Powers
Exponents and powers are essential concepts in algebra which simplify the process of working with repeated multiplication. An exponent indicates how many times a number, called the base, is multiplied by itself. For example, in the expression \( y^3 \), the base is \( y \) and the exponent is \( 3 \). This means \( y \) is multiplied by itself three times.

Calculating powers is straightforward once you understand the notation:
  • \( y^3 = y \times y \times y \).
  • Given \( y = 5 \), then \( y^3 = 5 \times 5 \times 5 = 125 \).
Working with exponents allows us to handle large numbers efficiently and accurately. It's a key skill that often overlaps with evaluating expressions and applying order of operations, which are discussed later.
Order of Operations
The order of operations is crucial for evaluating expressions correctly. It dictates the sequence in which different operations must be performed. The standard acronym PEMDAS can help you remember the order:
  • P: Parentheses
  • E: Exponents
  • M/D: Multiplication and Division (from left to right)
  • A/S: Addition and Subtraction (from left to right)
In the expression \( 5^3 - 4 \times 2 \), remember:
  • Evaluate exponents first, like \( 5^3 \).
  • Next, deal with multiplication \( 4 \times 2 \).
  • Finally, perform subtraction \( 125 - 8 \).
Mistake-free calculations rely on using this systematic approach, ensuring you evaluate the parts of the expression in the right order.
Evaluating Expressions
Evaluating an expression involves substituting variables with their given values and then simplifying the results using the order of operations. Here, we have the expression \( y^3 - 4x \), and we substitute \( y = 5 \) and \( x = 2 \). It transforms the problem to:
  • Calculate \( 5^3 \) yielding \( 125 \).
  • Find \( 4 \times 2 \) to get \( 8 \).
  • Subtract these results: \( 125 - 8 = 117 \).
By carefully substituting and evaluating step by step, you ensure accurate results without missing any calculations. This methodical approach emphasizes the importance of each mathematical concept, like exponents and the order of operations, in solving algebraic expressions accurately.

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