Solving And Simplifying Mathematical Expressions A Step By Step Guide

In this article, we will tackle several mathematical expressions, focusing on simplifying and solving them step by step. Mathematics is a fundamental tool for understanding the world around us, and mastering algebraic expressions is crucial for success in various fields. We will break down each expression, clarifying ambiguous notations and applying the correct order of operations. Our primary goal is to provide clear, understandable solutions that enhance your mathematical skills. By delving into these examples, you'll gain confidence in your ability to manipulate and solve complex expressions, which is a valuable skill in both academic and real-world scenarios. Let's embark on this mathematical journey together, unraveling the intricacies of each expression and solidifying your understanding of fundamental concepts.

1. Simplifying 6 * 3x * 6x * (5 + 4)

Let’s begin by simplifying the first expression: 6 * 3x * 6x * (5 + 4). This expression involves multiplication and addition, and we must adhere to the order of operations (PEMDAS/BODMAS) to arrive at the correct solution. The acronym PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This means we first address any operations within parentheses, then exponents, followed by multiplication and division, and finally addition and subtraction. In this specific case, we have parentheses, multiplication, and variables, making it a good example to illustrate the step-by-step simplification process. Understanding these steps is crucial for tackling more complex algebraic expressions in the future. So, let's meticulously walk through the process to ensure clarity and accuracy. We will start by simplifying the parentheses and then proceed with the multiplications, combining like terms as necessary to reach the final simplified form. This methodical approach will help solidify your understanding of algebraic manipulation.

Step-by-step Solution

1. Simplify inside the parentheses:

   • (5 + 4) = 9

2. Rewrite the expression:

   • 6 * 3x * 6x * 9

3. Multiply the constants:

   • 6 * 3 * 6 * 9 = 972

4. Multiply the variables:

   • x * x = x^2

5. Combine the constants and variables:

   • 972 * x^2 = 972x^2

Therefore, the simplified expression is 972x^2. This demonstrates how careful application of the order of operations and algebraic principles can simplify a seemingly complex expression into a much more manageable form. Remember, breaking down the problem into smaller, more manageable steps is often the key to success in mathematics.

2. Clarifying and Solving 7 C25 + 11 * 2

The expression 7 C25 + 11 * 2 presents an interesting challenge, particularly due to the notation “7 C25.” In mathematics, the notation “nCr” typically refers to combinations, specifically “n choose r,” which is the number of ways to choose r items from a set of n items without regard to order. This concept is fundamental in combinatorics, a branch of mathematics concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. If “7 C25” indeed represents a combination, it would be written as ₇C₂₅ or (7 choose 25). However, there's a critical issue here: in combinations, r must be less than or equal to n. In our case, 25 is greater than 7, which means ₇C₂₅ is not a standard combination and would typically be undefined in the context of basic combinatorics. Therefore, we need to consider other possible interpretations or correct the expression to make it mathematically valid.

Interpretation and Solution

Given the ambiguity, we'll proceed under two assumptions:

Assumption 1: Typographical Error

If “7 C25” is a typographical error and should be interpreted as a combination where the numbers are swapped, i.e., ₂₅C₇ (25 choose 7), we can calculate it. However, without further clarification, this is speculative. Alternatively, if the 'C' represents another mathematical operation or constant, that would drastically change the approach. For instance, 'C' could potentially be a constant or a coefficient in a specific context, but without additional information, this remains uncertain. Therefore, clarifying the intended meaning of 'C' is paramount before proceeding with a solution. Ambiguity in mathematical expressions can lead to incorrect results, so precision in notation is crucial. We'll first explore the scenario where a combination was intended, bearing in mind the limitations due to the numbers provided.

Assumption 2: 'C' is a Variable or Constant

If 'C' represents a variable or constant, we cannot simplify 7 C25 without knowing the value or operation it represents. This underscores the importance of clear notation in mathematics. Without a defined operation or value for 'C', the term 7 C25 remains undefined in its current form. This situation highlights how mathematical expressions are precise languages, and any deviation from standard notation or lack of clarity can impede the solving process.

Solving with a Corrected Combination

Assuming the intention was ₇C₂ (7 choose 2) or ₇C₅ (7 choose 5), we'll proceed with ₇C₂ for demonstration:

• Combination Formula: nCr = n! / (r! * (n - r)!) where