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Chapter 0: Problem 3

evaluate the expression for the given value of x. $$ 4 x^{-3} \quad x=2 $$

Short Answer

Expert verified
The value of the expression \(4x^{-3}\) for \(x=2\) is 0.5.

Step by step solution

01

Understand the meaning of negative exponent

The negative exponent in \(x^{-3}\) means 1 divided by \(x^3\). So, the original expression \(4x^{-3}\) can be rewritten as \(4 / (x^3)\).
02

Substitute the value of x

The next step is to replace x in the expression with the given value which is 2. Hence, the expression becomes \(4 / (2^3)\).
03

Simplify the expression

Last, do the math. \(2^3\) is 8, so you have \(4 / 8\), and that simplifies to 0.5.
04

Write down the answer

After evaluating the expression for \(x=2\), the final answer is 0.5.

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Key Concepts

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

Expression Evaluation
Expression evaluation is the process of finding the value of an expression for a given variable. Here, we are looking at the expression \(4 x^{-3}\) and need to evaluate it for \(x = 2\). Sometimes, expressions include components like negative exponents, terms that need to be carefully interpreted and computed. This requires understanding not just how to perform operations, but also interpreting each component accurately, especially with operations like division, multiplication, and exponentiation. By evaluating, we ensure that the expression is solved correctly, giving us a precise numerical value.
Substitution
Substitution is a crucial step in evaluating expressions. It involves replacing a variable in an expression with its given numerical value. This action is what allows us to transform an abstract expression into a concrete computational task.
In our specific problem, we're given that \(x=2\). So, we substitute 2 in place of \(x\) in the expression \(4 x^{-3}\). This substitution transforms the expression to \(4 / (2^3)\). Performing substitution accurately ensures the expression remains mathematically equivalent while making it easier to compute.
Simplifying Expressions
Simplifying expressions is about making them easier to compute or understand. After substitution, we need to simplify \(4 / (2^3)\).
- First, calculate \(2^3\) which results in 8.- Substitute back into the expression, making it \(4 / 8\). Simplifying helps us reduce complex expressions into more manageable forms. Whenever possible, we should simplify step-by-step and write down intermediary steps to avoid mistakes and make the process clear.
Mathematical Computation
Mathematical computation refers to the actual arithmetic operations carried out on the simplified expression. Now that we have \(4 / 8\), we perform the division operation:- Divide 4 by 8, resulting in 0.5.Effective computation requires careful handling of numbers and operations to ensure accuracy. Double-checking one's work at each step can help avoid small errors which may lead to incorrect results. Successfully performing these computations is the key to correctly evaluating expressions and arriving at the correct final answer.

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