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Chapter 2: Problem 12

Evaluate each expression when \(x=-3, y=-5,\) and \(z=0.\) $$y-x^{3}$$

Short Answer

Expert verified
The value of the expression \(y-x^{3}\) with \(x = -3\) and \(y = -5\) equals 22.

Step by step solution

01

Substitution

Substitute the given values into the equation \(y - x^{3}\). Thus, the equation becomes \(-5 - (-3)^{3}\).
02

Evaluate Exponent

First, solve the exponent. \((-3)^{3}\) equals -27.
03

Simplify and Resolve

Replace \(-3^{3}\) with -27 in the equation. The equation is now \(-5 - (-27)\), which simplifies to \(-5 + 27 = 22\).

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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 key technique in evaluating expressions, particularly in algebra. It involves replacing variables with numerical values to simplify the problem.
  • Identify the variables within the expression that need replacement. In our original exercise, these variables are \(x\) and \(y\).
  • Use the given values for substitution. For example, we are given \(x = -3\), \(y = -5\), and \(z = 0\).
  • Substitute these values into the expression to transform it. In this specific exercise, \(y - x^3\) becomes \(-5 - (-3)^3\) after substitution.
The substitution method allows us to transform an algebraic expression into an arithmetic one, which is easier to solve. After substitution, you're often left with an expression that involves numerical computations like exponentiation and basic arithmetic.
Exponents
Exponents are a powerful tool in algebra that represent repeated multiplication of a number by itself. Here, understanding how to manage negative bases and exponents is crucial. For example, when evaluating \((-3)^3\):
  • The base is \(-3\), and it is raised to the power of 3.
  • You multiply \(-3\) by itself three times: \(-3 \times -3 \times -3\).
  • Start by solving \(-3 \times -3\), which results in 9 because a negative times a negative is positive.
  • Then multiply 9 by -3, resulting in -27.
Exponents, especially with negative bases, can change the sign of the result depending on whether the exponent is even or odd. With odd exponents, like in this case, the result keeps the sign of the base (negative), resulting in -27 for \((-3)^3\).
Integer Arithmetic
Integer arithmetic involves additions, subtractions, multiplications, and divisions with whole numbers, including negatives. This skill is essential when simplifying expressions. Looking back at our example, after evaluating the exponent, we dealt with the operation \(-5 - (-27)\):
  • The subtraction of a negative number is equivalent to adding its positive counterpart. Thus, \(-5 - (-27)\) becomes \(-5 + 27\).
  • When adding numbers with different signs, find their absolute values: absolute value of -5 is 5, and of 27 is 27.
  • Subtract the smaller absolute value from the larger one: \(27 - 5 = 22\).
  • Use the sign of the number with the larger absolute value. Here, 27 is larger than 5, and it is positive, so the result is positive: 22.
Understanding how to handle negative numbers correctly is crucial for solving algebraic problems accurately. Integer arithmetic is fundamental in arriving at the final answer of an expression.

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