Evaluate Expression: S=2, T=-1.5
Let's dive into the world of algebraic expressions! In this article, we'll break down how to evaluate the expression -3|5-s³|+3t when given the values s=2 and t=-1.5. This type of problem is a fundamental concept in mathematics, often encountered in algebra and pre-calculus courses. Understanding the step-by-step process not only helps in solving similar problems but also builds a strong foundation for more advanced mathematical concepts. We will explore each component of the expression, from exponents and absolute values to multiplication and addition, ensuring a clear understanding of the solution. By the end of this article, you'll be equipped to tackle similar evaluation problems with confidence and precision.
Understanding the Expression
Before we jump into substituting the values, let's understand the expression itself. The expression -3|5-s³|+3t consists of several mathematical operations. It's crucial to break it down into smaller parts to avoid confusion. First, we have the term s³, which means s cubed or s raised to the power of 3. Then, we have |5-s³|, which represents the absolute value of the difference between 5 and s³. Remember, the absolute value of a number is its distance from zero, so it's always non-negative. Next, we have the multiplication of -3 with the absolute value and 3 with t. Finally, we have the addition of these two products. Understanding the order of operations, often remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction), is crucial here. This order dictates the sequence in which we perform the operations to arrive at the correct answer. Grasping these individual components and the order in which they interact is the first step towards successfully evaluating the expression.
Step-by-Step Evaluation
Now, let's proceed with the step-by-step evaluation of the expression. This is where we substitute the given values of s and t into the expression and simplify. Our expression is -3|5-s³|+3t, and we are given s=2 and t=-1.5. The first step is to substitute these values into the expression. Replacing s with 2 and t with -1.5, we get -3|5-2³|+3(-1.5). Now, we need to follow the order of operations (PEMDAS/BODMAS). According to this order, we first deal with the exponent. 2³ is 2 multiplied by itself three times, which equals 8. So, our expression becomes -3|5-8|+3(-1.5). Next, we handle the operation inside the absolute value. 5 minus 8 is -3, so we have -3|-3|+3(-1.5). The absolute value of -3 is 3, as absolute value is the distance from zero. Our expression now simplifies to -3(3)+3(-1.5). Next, we perform the multiplications. -3 multiplied by 3 is -9, and 3 multiplied by -1.5 is -4.5. So, the expression becomes -9 + (-4.5). Finally, we perform the addition. -9 plus -4.5 equals -13.5. Therefore, the value of the expression -3|5-s³|+3t when s=2 and t=-1.5 is -13.5. Each step is vital in reaching the correct result, emphasizing the importance of accuracy in substitution and following the order of operations.
Detailed Calculation Breakdown
To further solidify our understanding, let's break down the calculation process in even greater detail. We started with the expression -3|5-s³|+3t and the values s=2 and t=-1.5. First, we substituted these values into the expression, resulting in -3|5-2³|+3(-1.5). The next critical step was evaluating the exponent. 2 cubed (2³) is 222, which equals 8. Substituting this back into our expression gives us -3|5-8|+3(-1.5). Then, we focused on the operation within the absolute value bars. 5 minus 8 is -3, so we now have -3|-3|+3(-1.5). The absolute value of -3, denoted as |-3|, is the non-negative value of -3, which is 3. Thus, the expression becomes -3(3)+3(-1.5). Now, we move on to the multiplication operations. -3 multiplied by 3 is -9, and 3 multiplied by -1.5 is -4.5. Our expression is now -9 + (-4.5). Finally, we perform the addition: -9 plus -4.5. When adding two negative numbers, we add their absolute values and keep the negative sign. So, 9 + 4.5 is 13.5, and since both numbers were negative, the result is -13.5. Therefore, the final value of the expression when s=2 and t=-1.5 is -13.5. This detailed breakdown emphasizes the significance of each step, ensuring a clear understanding of how we arrive at the solution.
Common Mistakes to Avoid
When evaluating algebraic expressions, especially those involving absolute values and exponents, it's easy to make mistakes. Recognizing these common pitfalls can significantly improve accuracy. One frequent error is overlooking the order of operations (PEMDAS/BODMAS). For instance, calculating 5-2³ as (5-2)³ instead of 5-(2³) can lead to an incorrect result. Always prioritize exponents before subtraction. Another common mistake involves the absolute value. Remember that the absolute value of a number is its distance from zero, always a non-negative value. So, |-3| is 3, not -3. Forgetting this can change the entire outcome of the expression. Sign errors are also prevalent, particularly when dealing with negative numbers. Ensure careful handling of negative signs during multiplication and addition. For example, -3 * -1.5 is positive 4.5, not negative. Substituting values correctly is also crucial. Double-check that you've replaced each variable with its corresponding value accurately. A simple substitution error at the beginning can cascade through the entire calculation. Lastly, breaking the problem into smaller, manageable steps helps avoid errors. Trying to do too much at once can lead to confusion and mistakes. By being aware of these common errors and taking a methodical approach, you can greatly increase your chances of arriving at the correct solution.
Practice Problems and Further Learning
To truly master evaluating algebraic expressions, practice is key. Working through various problems helps solidify your understanding and build confidence. Try evaluating similar expressions with different values for s and t. For example, you could try s=-1 and t=2, or even more complex values like fractions or decimals. You can also vary the expression itself by adding more terms, exponents, or absolute value components. Online resources, textbooks, and worksheets offer a plethora of practice problems with varying levels of difficulty. Don't hesitate to seek out additional learning materials and tutorials. Websites like Khan Academy offer excellent videos and exercises on algebraic expressions and the order of operations. Engaging with different types of problems and resources will deepen your understanding and sharpen your skills. Furthermore, understanding the underlying principles of algebra, such as the commutative, associative, and distributive properties, can provide a broader context for evaluating expressions. The more you practice and explore, the more comfortable and proficient you'll become in handling these types of mathematical problems. Remember, consistency and persistence are essential for success in mathematics. Regularly practicing and seeking out new challenges will help you develop a strong foundation in algebra and beyond.
In conclusion, we've successfully evaluated the expression -3|5-s³|+3t when s=2 and t=-1.5. We've broken down the expression, followed the order of operations, and discussed common mistakes to avoid. Remember, the key to success in mathematics is understanding the underlying concepts and consistent practice. For further learning and to deepen your understanding of algebraic expressions, visit Khan Academy's Algebra I course.
