Simplifying Complex Math Expression A Step-by-Step Guide

Mathematical expressions, especially those involving radicals, exponents, and various arithmetic operations, can often appear daunting at first glance. However, by systematically applying the order of operations and leveraging key mathematical principles, we can break down even the most intricate expressions into manageable steps. In this article, we will delve into a complex mathematical expression, unraveling its layers to arrive at a simplified solution. Our focus will be on providing a clear, step-by-step explanation, ensuring that each operation is thoroughly justified. This approach not only aids in understanding the solution but also equips you with the skills to tackle similar mathematical challenges with confidence.

To begin, let's consider the expression:

$$\frac{(-\sqrt{25})^3 \times(\sqrt[3]{-8})^2}{-3(-3)^2+2 \times 2+3}$$

This expression encompasses a variety of mathematical operations, including square roots, cube roots, exponents, multiplication, and addition. The numerator involves the product of two terms, each of which requires simplification. The denominator, on the other hand, consists of a series of arithmetic operations that must be performed in the correct order. Before we dive into the detailed steps, it's essential to understand the underlying principles that govern the simplification process. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which operations must be performed to ensure a consistent and accurate result. Additionally, understanding the properties of exponents and radicals is crucial for simplifying the terms within the expression.

Breaking Down the Numerator

The numerator of our expression is $(-\sqrt{25})^3 \times(\sqrt[3]{-8})^2$. To simplify this, we'll address each part separately before combining them. Let's start with the first term, $(-\sqrt{25})^3$. The square root of 25 is 5, so we have $(-\5)^3$. This means -5 multiplied by itself three times: $(-5) \times (-5) \times (-5)$. When we multiply the first two -5s, we get 25. Then, multiplying 25 by -5 gives us -125. So, $(-\sqrt{25})^3$ simplifies to -125. This initial step demonstrates the importance of addressing radicals and exponents before moving on to multiplication. The square root operation effectively reduces the radical term to a simple integer, while the exponentiation calculates the power of the resulting value. These operations are foundational to simplifying more complex expressions and require a solid understanding to execute accurately.

Moving on to the second term in the numerator, we have $(\sqrt[3]{-8})^2$. The cube root of -8 is -2 because $(-2) \times (-2) \times (-2) = -8$. So, we have $(-2)^2$, which means -2 multiplied by itself: $(-2) \times (-2)$. This equals 4. Therefore, $(\sqrt[3]{-8})^2$ simplifies to 4. The process of finding the cube root of a negative number is a critical step here. Unlike square roots, cube roots can yield negative results, as demonstrated by the cube root of -8 being -2. This understanding is crucial for handling expressions involving radicals of odd indices. Squaring the result then transforms the negative value into a positive one, highlighting the importance of paying close attention to the order of operations and the properties of exponents.

Now that we've simplified both parts of the numerator, we can multiply them together: -125 multiplied by 4. This gives us -500. Therefore, the entire numerator simplifies to -500. This final step in simplifying the numerator combines the results of the previous operations. By breaking down the initial expression into smaller, more manageable parts, we've systematically simplified the numerator to a single numerical value. This approach underscores the power of modular simplification in handling complex mathematical expressions. Each step builds upon the previous one, ultimately leading to a concise and accurate result.

Simplifying the Denominator

Now, let’s turn our attention to the denominator of the expression: $-3(-3)^2+2 \times 2+3$. According to the order of operations (PEMDAS), we need to address exponents first. We have $(-3)^2$, which is -3 multiplied by itself: $(-3) \times (-3)$, resulting in 9. So, the expression becomes $-3(9)+2 \times 2+3$. This initial step in simplifying the denominator highlights the critical importance of adhering to the order of operations. Exponents take precedence over multiplication and addition, and failing to recognize this can lead to incorrect results. By correctly evaluating the exponent first, we set the stage for the subsequent operations, ensuring that the simplification process remains accurate.

Next, we perform the multiplication operations. We have $-3 \times 9$, which is -27, and $2 \times 2$, which is 4. So, the expression now looks like this: $-27+4+3$. This step involves performing multiplication before addition, as dictated by the order of operations. Each multiplication is carried out independently, transforming the expression into a series of addition and subtraction operations. This transition is crucial for moving towards the final simplified form, as addition and subtraction are the final steps in the order of operations.

Finally, we perform the addition and subtraction from left to right. First, we add -27 and 4, which gives us -23. Then, we add -23 and 3, which results in -20. Therefore, the denominator simplifies to -20. The final step in simplifying the denominator involves combining the remaining terms through addition and subtraction. By performing these operations from left to right, we ensure that the order of operations is strictly followed. The result, -20, represents the fully simplified form of the denominator, ready to be used in the final calculation of the entire expression.

Combining Numerator and Denominator

Now that we've simplified both the numerator and the denominator, we can combine them to find the final value of the expression. We have the numerator simplified to -500 and the denominator simplified to -20. So, the expression becomes $\frac{-500}{-20}$. To simplify this fraction, we divide -500 by -20. A negative number divided by a negative number results in a positive number. So, we have $500 \div 20$. This division gives us 25. Therefore, the final simplified value of the expression is 25. This final step combines the results of all the previous simplifications. By dividing the simplified numerator by the simplified denominator, we arrive at the ultimate solution. The fact that the result is a positive integer is a testament to the accuracy of each preceding step. This final answer underscores the power of systematic simplification and the importance of a thorough understanding of mathematical principles.

Conclusion

In this article, we've systematically simplified the complex mathematical expression $\frac{(-\sqrt{25})^3 \times(\sqrt[3]{-8})^2}{-3(-3)^2+2 \times 2+3}$. By breaking down the expression into smaller, more manageable parts, we were able to apply the order of operations and key mathematical principles to arrive at the final answer of 25. This process demonstrates the importance of a step-by-step approach in tackling complex mathematical problems. Each operation, from simplifying radicals and exponents to performing multiplication, division, addition, and subtraction, was carefully executed to ensure accuracy. The methodical simplification of the numerator and denominator separately, followed by their combination, highlights the effectiveness of modular problem-solving techniques. The entire process not only provides the solution but also offers a clear understanding of the underlying mathematical concepts. By mastering these techniques, you can confidently approach and solve a wide range of mathematical expressions.

Through this detailed exploration, we hope to have illuminated the path to simplifying complex mathematical expressions. Remember, the key to success lies in understanding the fundamental principles, applying the order of operations diligently, and breaking down problems into manageable steps. With practice and perseverance, you can unlock the beauty and power of mathematics.