Simplify Expressions A Step By Step Guide

When simplifying mathematical expressions, it's vital to follow a systematic approach to ensure accuracy and clarity. In this section, we will delve into the simplification of the expression $9 \frac{(5+6)^2}{5+3x-6}$. Simplifying this expression involves a series of steps, including performing arithmetic operations, combining like terms, and reducing the expression to its simplest form. We'll break down each step in detail to provide a clear understanding of the process.

First, let's address the numerator of the fraction. The term $(5+6)^2$ requires us to first perform the addition inside the parentheses and then square the result. $5+6$ equals $11$, so we have $11^2$, which is $11 * 11 = 121$. Now, we multiply this result by 9, giving us $9 * 121 = 1089$. Therefore, the numerator of the expression simplifies to 1089.

Next, we turn our attention to the denominator, which is $5+3x-6$. Simplifying the denominator involves combining the constant terms. We have $5 - 6$, which equals $-1$. Thus, the denominator becomes $3x - 1$. At this stage, the expression looks like this: $\frac{1089}{3x-1}$. This is the simplified form of the original expression, assuming that the denominator cannot be further factored or simplified.

It is important to note that we should also consider any restrictions on the variable $x$. The denominator cannot be equal to zero, as division by zero is undefined. So, we need to find the value(s) of $x$ that make $3x - 1 = 0$. Solving this equation, we add 1 to both sides to get $3x = 1$, and then divide by 3 to find $x = \frac{1}{3}$. Therefore, $x$ cannot be equal to $\frac{1}{3}$.

In summary, the simplified form of the expression $9 \frac{(5+6)^2}{5+3x-6}$ is $\frac{1089}{3x-1}$, with the restriction that $x \neq \frac{1}{3}$. This comprehensive walkthrough highlights the step-by-step process of simplifying complex expressions, emphasizing the importance of order of operations and identifying restrictions on variables. This approach to simplification is crucial for mastering algebraic manipulations and ensuring accurate mathematical solutions.

Now, let's tackle the simplification of the second expression: $\frac{x^2-8x+16}{x^2-2x-8}$. Simplifying rational expressions like this requires factoring both the numerator and the denominator and then canceling out any common factors. This process allows us to reduce the expression to its simplest form, making it easier to work with in further calculations or analyses.

We start by factoring the numerator, which is a quadratic expression: $x^2 - 8x + 16$. To factor this, we look for two numbers that multiply to 16 and add up to -8. Those numbers are -4 and -4. Therefore, the numerator can be factored as $(x - 4)(x - 4)$, which can also be written as $(x - 4)^2$.

Next, we factor the denominator, which is another quadratic expression: $x^2 - 2x - 8$. Factoring this quadratic involves finding two numbers that multiply to -8 and add up to -2. Those numbers are -4 and 2. Thus, the denominator can be factored as $(x - 4)(x + 2)$. Now, our expression looks like this: $\frac{(x-4)(x-4)}{(x-4)(x+2)}$.

Now we can cancel out the common factors. We see that $(x - 4)$ appears in both the numerator and the denominator, so we can cancel one instance of $(x - 4)$ from each. This leaves us with $\frac{x-4}{x+2}$. This is the simplified form of the original expression. However, we must also consider the restrictions on the variable $x$. The original denominator, $x^2 - 2x - 8$, cannot be equal to zero, as this would make the expression undefined. We factored the denominator as $(x - 4)(x + 2)$, so we need to find the values of $x$ that make these factors equal to zero.

Setting $x - 4 = 0$, we find $x = 4$. Setting $x + 2 = 0$, we find $x = -2$. Therefore, $x$ cannot be equal to 4 or -2. These are the restrictions on the variable $x$. In summary, the simplified form of the expression $\frac{x^2-8x+16}{x^2-2x-8}$ is $\frac{x-4}{x+2}$, with the restrictions that $x \neq 4$ and $x \neq -2$. This detailed breakdown illustrates the process of factoring quadratic expressions and identifying restrictions, which are essential skills in algebraic simplification. Mastering these techniques ensures accuracy and a thorough understanding of the properties of rational expressions.

In conclusion, the process of simplifying expressions is a cornerstone of algebra and essential for solving more complex mathematical problems. We've explored two distinct examples, each illustrating different techniques and considerations. The first example, $9 \frac{(5+6)^2}{5+3x-6}$, showcased the importance of the order of operations and combining like terms. We meticulously worked through each step, from evaluating the exponent to simplifying the denominator, ultimately arriving at the simplified form $\frac{1089}{3x-1}$, with the crucial caveat that $x \neq \frac{1}{3}$. This thorough approach highlights the necessity of identifying and stating restrictions on variables to ensure mathematical rigor.

The second example, $\frac{x^2-8x+16}{x^2-2x-8}$, delved into the realm of rational expressions, emphasizing the critical skill of factoring. We demonstrated how to factor both the numerator and the denominator, identify common factors, and cancel them to reduce the expression to its simplest form. Factoring quadratics is a fundamental technique in algebra, and this example underscores its significance in simplifying complex fractions. The simplified form, $\frac{x-4}{x+2}$, was accompanied by the restrictions $x \neq 4$ and $x \neq -2$, reinforcing the need to consider values that would make the original denominator zero.

Mastering the art of simplification not only enhances problem-solving skills but also provides a deeper understanding of mathematical structures and relationships. By breaking down complex expressions into their simplest forms, we gain clarity and insight, making it easier to analyze and manipulate equations. The ability to simplify expressions is invaluable in various fields, from engineering and physics to economics and computer science. Whether it's reducing a complex formula to a manageable equation or identifying key factors in a mathematical model, simplification is a powerful tool for both students and professionals.

Furthermore, the process of simplification promotes logical thinking and attention to detail. Each step requires careful consideration and precision, fostering a methodical approach to problem-solving. This meticulousness is not only beneficial in mathematics but also in any discipline that demands analytical skills. By consistently practicing and refining simplification techniques, individuals can develop a strong foundation for tackling more advanced mathematical concepts and real-world challenges.

In summary, simplifying expressions is more than just a mathematical exercise; it's a fundamental skill that enhances understanding, promotes logical thinking, and empowers individuals to solve complex problems effectively. The techniques and principles discussed in this guide provide a solid foundation for mastering this essential skill and applying it across various disciplines.