How to Simplify Expressions with Absolute Values: A Step-by-Step Guide
Quick Answer
To simplify the expression | -1 |^3 - 9x - 4, first find the absolute value, then cube it, and combine like terms. The final expression is -3 - 9x.
Understanding how to simplify expressions that involve absolute values is a crucial skill in mathematics. Let’s break down the expression | -1 |^3 - 9x - 4 step by step to clarify this process.
1. **Absolute Value Calculation**: The first step is to calculate the absolute value of -1. The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, | -1 | equals 1.
2. **Cubing the Result**: Next, we take that result and raise it to the power of 3. So, we compute 1^3, which equals 1.
3. **Revising the Expression**: With the absolute value simplified, we can now rewrite our original expression. We replace | -1 |^3 with 1, giving us the new expression: 1 - 9x - 4.
4. **Combining Like Terms**: Now, we need to combine like terms. We can simplify 1 - 4, which equals -3. Our expression now reads: -3 - 9x.
5. **Final Transformation**: If preferred, this can also be expressed as -(9x + 3), which is a common technique used in algebra to factor out a negative sign.
### Real-World Applications
Understanding how to work with absolute values and algebraic expressions is not just an academic exercise; it has real-world applications. For instance, in fields like engineering, physics, and economics, the ability to simplify expressions can help in designing structures, calculating forces, or analyzing financial models. Furthermore, mastering these concepts lays a strong foundation for more advanced topics in mathematics, such as calculus and statistics.
### Example Problem
Let’s consider another example: Simplifying | 5 |^2 - 3x + 2. First, we find the absolute value: | 5 | = 5. Next, we square it: 5^2 = 25. Now, we replace and simplify: 25 - 3x + 2 = 27 - 3x. This process illustrates the same principles we used in the original expression.
### Conclusion
By breaking down and simplifying expressions step by step, you’ll develop a stronger understanding of algebra. This skill will not only aid in your current studies but also prepare you for future mathematical challenges. If you have further questions or need more examples, don’t hesitate to explore additional resources or ask for help!
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