Evaluate The Expression A Y^2 - Y^3 For A = 8.8 And Y = -1.2
Introduction
In this article, we will delve into the process of evaluating the algebraic expression a y^2 - y^3 given specific values for the variables a and y. This is a fundamental skill in algebra, often encountered in various mathematical contexts. Understanding how to substitute values into expressions and simplify them is crucial for solving equations, graphing functions, and tackling more complex algebraic problems. The given expression involves variables raised to powers, which necessitates a firm grasp of exponent rules and the order of operations. Our task is to replace a with 8.8 and y with -1.2, and then meticulously perform the arithmetic operations to arrive at the final numerical value of the expression. This exercise not only reinforces algebraic manipulation skills but also highlights the importance of precision in calculations, especially when dealing with negative numbers and decimals. The steps involved in this evaluation process provide a clear demonstration of how algebraic expressions are used to represent real-world relationships and how their values change depending on the values assigned to the variables. We will break down each step in detail, ensuring clarity and understanding for readers of all mathematical backgrounds.
Understanding the Expression
The algebraic expression we are tasked with evaluating is a y^2 - y^3. This expression consists of two terms: a y^2 and y^3. The first term, a y^2, involves the product of the variable a and the square of the variable y. This means we need to multiply a by y raised to the power of 2. The second term, y^3, represents y raised to the power of 3, which means we need to multiply y by itself three times. The minus sign between the two terms indicates that we will subtract the value of y^3 from the value of a y^2. Understanding the structure of the expression is the first step in correctly evaluating it. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which we perform the calculations. In this case, we will first deal with the exponents, then perform the multiplication, and finally carry out the subtraction. This careful approach ensures that we arrive at the correct value of the expression. Moreover, it's essential to pay close attention to the signs of the numbers, especially when dealing with negative values. A misplaced negative sign can lead to a significant error in the final result. Therefore, a methodical and step-by-step approach is crucial for accurately evaluating this expression.
Substituting the Values
The next step in evaluating the expression a y^2 - y^3 is to substitute the given values for the variables a and y. We are given that a = 8.8 and y = -1.2. This means we will replace every instance of a in the expression with 8.8 and every instance of y with -1.2. The expression then becomes: 8. 8 * (-1.2)^2 - (-1.2)^3. It's crucial to enclose the negative value of y in parentheses to ensure that the exponent applies correctly. This is particularly important when dealing with even exponents, as the square of a negative number is positive, while the cube of a negative number is negative. The substitution process transforms the algebraic expression into a numerical expression, which we can then simplify using the order of operations. This step is a fundamental aspect of algebra, as it allows us to determine the value of an expression for specific inputs. The accuracy of the substitution is paramount, as any error at this stage will propagate through the rest of the calculation. Therefore, it is good practice to double-check the substitution to ensure that the values have been correctly placed. This careful attention to detail is essential for achieving a correct final answer.
Calculating y^2
Now that we have substituted the values, our expression is 8.8 * (-1.2)^2 - (-1.2)^3. Following the order of operations (PEMDAS), we need to address the exponents first. Let's start by calculating y^2, which is (-1.2)^2. This means we need to multiply -1.2 by itself: (-1.2) * (-1.2). When multiplying two negative numbers, the result is positive. So, we have 1.2 * 1.2. To perform this multiplication, we can either do it manually or use a calculator. Manually, we can multiply 12 by 12, which gives us 144. Since we are multiplying two numbers with one decimal place each, the result will have two decimal places. Therefore, 1.2 * 1.2 = 1.44. So, (-1.2)^2 = 1.44. This calculation is a crucial step in evaluating the expression, as it provides the value that will be multiplied by a in the first term. Understanding how to handle negative numbers and exponents is essential for accurate algebraic manipulation. The positive result of squaring a negative number is a common concept in algebra, and it's important to remember this rule to avoid errors. With this value calculated, we can now move on to the next exponent in the expression.
Calculating y^3
Next, we need to calculate y^3, which is (-1.2)^3. This means we need to multiply -1.2 by itself three times: (-1.2) * (-1.2) * (-1.2). We already know that (-1.2) * (-1.2) = 1.44 from the previous step. So, we now need to multiply 1.44 by -1.2. When multiplying a positive number by a negative number, the result is negative. So, we will have a negative result. Now, let's multiply 1.44 by 1.2. Again, we can do this manually or use a calculator. Manually, we can multiply 144 by 12, which gives us 1728. Since we are multiplying a number with two decimal places by a number with one decimal place, the result will have three decimal places. Therefore, 1.44 * 1.2 = 1.728. So, (-1.2)^3 = -1.728. This calculation is another critical step in evaluating the expression, as it provides the value of the second term that will be subtracted from the first term. The negative result of cubing a negative number is a fundamental concept in algebra, and it's important to remember this rule. With this value calculated, we have now addressed all the exponents in the expression and can move on to the multiplication step.
Calculating a * y^2
Now that we have calculated y^2, which is 1.44, we can proceed to calculate the first term of the expression, a y^2. We know that a = 8.8, so we need to multiply 8.8 by 1.44. This means we need to calculate 8.8 * 1.44. We can perform this multiplication manually or use a calculator. Manually, we can multiply 88 by 144, which gives us 12672. Since we are multiplying a number with one decimal place by a number with two decimal places, the result will have three decimal places. Therefore, 8.8 * 1.44 = 12.672. So, a y^2 = 12.672. This calculation is an essential step in evaluating the expression, as it determines the value of the first term before we perform the subtraction. The accurate multiplication of decimals is a crucial skill in algebra, and it's important to pay attention to the placement of the decimal point in the result. With this value calculated, we now have all the individual components needed to complete the evaluation of the expression.
Final Calculation
We have now calculated all the individual parts of the expression a y^2 - y^3. We found that a y^2 = 12.672 and y^3 = -1.728. So, our expression now looks like this: 12. 672 - (-1.728). Remember that subtracting a negative number is the same as adding its positive counterpart. So, we can rewrite the expression as: 12. 672 + 1.728. Now, we simply need to add these two numbers together. We can do this manually or use a calculator. Manually, we can align the decimal points and add the numbers column by column:
12.672
+ 1.728
--------
14.400
So, 12.672 + 1.728 = 14.4. Therefore, the value of the expression a y^2 - y^3 for a = 8.8 and y = -1.2 is 14.4. This final calculation combines all the previous steps to arrive at the solution. The careful application of the order of operations, the correct handling of negative numbers and decimals, and the accurate execution of each arithmetic operation are all crucial for obtaining the correct answer. This comprehensive evaluation demonstrates the power of algebraic manipulation in determining the value of an expression for specific variable values.
Conclusion
In conclusion, we have successfully evaluated the expression a y^2 - y^3 for a = 8.8 and y = -1.2. By systematically substituting the given values, calculating the exponents, performing the multiplication, and finally carrying out the subtraction, we arrived at the final value of 14.4. This exercise underscores the importance of understanding and applying the order of operations, as well as the rules for working with negative numbers and decimals. Each step in the process, from the initial substitution to the final addition, requires careful attention to detail to avoid errors. This type of algebraic evaluation is a fundamental skill in mathematics and is essential for solving more complex problems in various fields, including science, engineering, and economics. The ability to accurately manipulate expressions and equations is a cornerstone of mathematical literacy. Therefore, mastering these basic algebraic techniques is crucial for anyone pursuing further studies or careers in mathematically intensive disciplines. The step-by-step approach demonstrated in this article provides a clear and concise method for evaluating similar expressions, ensuring a solid understanding of the underlying principles.
