Simplifying And Evaluating Algebraic Expressions A Step By Step Guide
Introduction
In this article, we will walk through the process of simplifying and evaluating a complex algebraic expression. The expression we are dealing with is: $5 x y^2-\left{2 x^2 y-\left[3 x y^2-\left[2\left(2 x y2-x2 y\right)\right]\right]\right}$. We are given the values of the variables $x$ and $y$ as $x=-\frac{1}{3}$ and $y=-\frac{1}{2}$. Our goal is to simplify the expression first and then substitute these values to find the final numerical result. This involves using the order of operations, combining like terms, and careful arithmetic. This article provides a detailed, step-by-step guide to solving this problem, ensuring clarity and understanding for readers of all backgrounds. Algebraic simplification is a fundamental skill in mathematics, and mastering it is crucial for solving more complex problems in various fields, including physics, engineering, and computer science.
Step 1: Simplify the Innermost Parentheses
We begin by focusing on the innermost part of the expression, which is the term inside the parentheses: $2(2xy^2 - x^2y)$. To simplify this, we distribute the 2 across the terms inside the parentheses. This means we multiply each term inside the parentheses by 2. Doing so, we get: $2 * (2xy^2) - 2 * (x^2y) = 4xy^2 - 2x^2y$. This simplification is a crucial first step because it clears the way for us to tackle the outer layers of brackets and braces. By addressing the innermost terms first, we reduce the complexity of the overall expression, making it easier to manage and reducing the likelihood of errors. This process is in line with the standard order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), which dictates the sequence in which mathematical operations should be performed. Understanding and applying this order is essential for accurately simplifying algebraic expressions. The ability to correctly distribute and combine like terms is a foundational skill in algebra, which is used to solve equations, inequalities, and a myriad of other mathematical problems. Mastering these skills enables us to solve more complex problems with confidence and precision.
Step 2: Simplify the Square Brackets
Now that we've simplified the innermost parentheses, we move our attention to the square brackets that contain the expression $3xy^2 - [2(2xy^2 - x^2y)]$. We've already simplified the term inside the parentheses as $4xy^2 - 2x^2y$, so we can substitute this back into the expression within the square brackets. This gives us $3xy^2 - [4xy^2 - 2x^2y]$. Next, we need to eliminate the square brackets. We do this by distributing the negative sign (which is equivalent to multiplying by -1) across the terms inside the brackets. This means we change the sign of each term: $-1 * (4xy^2) = -4xy^2$ and $-1 * (-2x^2y) = +2x^2y$. Combining these results, we can rewrite the expression inside the square brackets as $3xy^2 - 4xy^2 + 2x^2y$. Now, we look for like terms that can be combined. In this case, we have $3xy^2$ and $-4xy^2$, which are like terms because they both contain the same variables raised to the same powers. Combining these, we get $(3 - 4)xy^2 = -1xy^2$, which we can write simply as $-xy^2$. Thus, our expression within the square brackets simplifies to $-xy^2 + 2x^2y$. By following this methodical approach, we continue to reduce the complexity of the expression one step at a time. This step is pivotal because it not only simplifies a significant portion of the original expression but also prepares it for the next stage of simplification, where we deal with the curly braces. Accurately handling negative signs and combining like terms are crucial skills in algebra, and these steps provide valuable practice in mastering these concepts.
Step 3: Simplify the Curly Braces
With the square brackets simplified, our next task is to tackle the curly braces. The expression within the curly braces is $2x^2y - [3xy^2 - [2(2xy^2 - x^2y)]]$. From the previous step, we know that the expression within the square brackets simplifies to $-xy^2 + 2x^2y$. Substituting this back into the curly braces, we get $2x^2y - (-xy^2 + 2x^2y)$. To eliminate the curly braces, we need to distribute the negative sign across the terms inside. This means multiplying each term inside the parentheses by -1. Doing so, we have $-1 * (-xy^2) = +xy^2$ and $-1 * (2x^2y) = -2x^2y$. So, the expression inside the curly braces becomes $2x^2y + xy^2 - 2x^2y$. Now, we identify and combine like terms. In this case, we have $2x^2y$ and $-2x^2y$, which are like terms. Combining these, we get $(2 - 2)x^2y = 0x^2y = 0$. This simplifies the expression within the curly braces to just $xy^2$. This step demonstrates the power of methodical simplification, as what initially appeared to be a complex expression has been significantly reduced. By consistently applying the principles of algebra, we've navigated through multiple layers of brackets and parentheses, ultimately simplifying the expression to a much more manageable form. This methodical approach not only makes the problem more tractable but also reduces the chances of making errors along the way. The simplification within the curly braces is a crucial milestone in the overall process, as it sets the stage for the final simplification and evaluation of the entire expression.
Step 4: Simplify the Entire Expression
Now that we've simplified the innermost parentheses, square brackets, and curly braces, we can simplify the entire expression. The original expression is $5xy^2 - {2x^2y - [3xy^2 - [2(2xy^2 - x^2y)]]}$. We've simplified the portion within the curly braces to $xy^2$, so we substitute this back into the original expression. This gives us $5xy^2 - xy^2$. This step involves combining like terms. Here, $5xy^2$ and $-xy^2$ are like terms because they both have the same variables raised to the same powers. To combine them, we subtract the coefficients: $5 - 1 = 4$. Therefore, the simplified expression is $4xy^2$. This result is a testament to the power of simplification. By systematically working through the expression, addressing each layer of complexity, we've transformed a seemingly complicated problem into a straightforward expression. The simplification process not only makes the expression easier to understand but also sets the stage for the final evaluation, where we substitute the given values for $x$ and $y$ to find a numerical answer. This step is crucial because it encapsulates all the previous simplifications, resulting in a compact and manageable expression that is ready for the final calculation. The ability to simplify expressions is a cornerstone of algebraic manipulation, enabling us to solve complex equations and problems with greater ease and accuracy.
Step 5: Substitute the Values of x and y
Having simplified the expression to $4xy^2$, we are now ready to substitute the given values of $x$ and $y$. We are given that $x = -\frac1}{3}$ and $y = -\frac{1}{2}$. Substituting these values into the simplified expression, we get3}) * (-\frac{1}{2})^2$. The first step is to evaluate the exponent. We have $(-\frac{1}{2})^2$, which means $(-\frac{1}{2}) * (-\frac{1}{2}) = \frac{1}{4}$. Now we substitute this value back into the expression3}) * (\frac{1}{4})$. Next, we perform the multiplication. We have3}) = -\frac{4}{3}$. Then we multiply this result by $\frac{1}{4}$3}) * (\frac{1}{4})$. When multiplying fractions, we multiply the numerators and the denominators3 * 4} = \frac{-4}{12}$. Finally, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4{12 ÷ 4} = \frac{-1}{3}$. Therefore, the final value of the expression is $-\frac{1}{3}$. This final step demonstrates the importance of accurate arithmetic in mathematical problem-solving. After simplifying the expression, the substitution and evaluation process requires careful attention to detail to ensure the correct result is obtained. The ability to work with fractions, exponents, and negative numbers is crucial in this context. The final answer, $-\frac{1}{3}$, represents the numerical value of the original complex algebraic expression when $x = -\frac{1}{3}$ and $y = -\frac{1}{2}$. This comprehensive process, from simplifying the expression to substituting values, highlights the interconnectedness of algebraic and arithmetic skills in solving mathematical problems.
Final Answer
After systematically simplifying the expression and substituting the given values for $x$ and $y$, we have arrived at the final answer. The simplified expression $4xy^2$ evaluated at $x = -\frac1}{3}$ and $y = -\frac{1}{2}$ yields a result of $-\frac{1}{3}$. This process involved several key steps{3}$, represents the numerical value of the original complex expression under the given conditions. This exercise illustrates the importance of a methodical approach in mathematics. By breaking down a complex problem into smaller, manageable steps, we can navigate through the intricacies and arrive at the correct solution. The ability to simplify algebraic expressions and evaluate them is a fundamental skill in mathematics, with applications in various fields, including engineering, physics, and computer science. The final answer not only provides a numerical solution but also underscores the power of algebraic manipulation and the importance of precision in mathematical calculations.
