Simplifying Algebraic Expressions Step-by-Step Guide

by Scholario Team 53 views

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. This article provides a comprehensive guide on how to simplify complex algebraic expressions, focusing on the step-by-step process of multiplying and reducing fractions with variables and exponents. We will dissect a specific problem to illustrate these principles, ensuring clarity and understanding for learners of all levels. This guide aims to empower you with the knowledge and techniques needed to tackle similar problems with confidence and precision. Mastery of these concepts not only enhances your mathematical prowess but also lays a strong foundation for more advanced topics in algebra and calculus. Let's embark on this mathematical journey together and unravel the intricacies of simplifying algebraic expressions.

Problem Statement

Let's consider the expression:

4x426y5×22y34x3\frac{4 x^4}{26 y^5} \times \frac{22 y^3}{4 x^3}

Our goal is to simplify this expression to its simplest form. This involves multiplying the two fractions and then reducing the resulting fraction by canceling out common factors. The process requires a solid understanding of fraction multiplication, exponent rules, and simplification techniques. We will break down each step in detail to ensure clarity and comprehension. The ultimate aim is to present the expression in a form that is both concise and mathematically equivalent to the original, making it easier to work with in subsequent calculations or analyses. This skill is crucial in various mathematical contexts, from solving equations to evaluating complex functions. Therefore, mastering this simplification process is a valuable asset in your mathematical toolkit.

Step-by-Step Solution

1. Multiplying the Fractions

To multiply fractions, we multiply the numerators together and the denominators together:

4x426y5×22y34x3=4x4×22y326y5×4x3\frac{4 x^4}{26 y^5} \times \frac{22 y^3}{4 x^3} = \frac{4 x^4 \times 22 y^3}{26 y^5 \times 4 x^3}

This step involves a straightforward application of the rule for multiplying fractions. We combine the coefficients, variables, and exponents in the numerators and denominators separately. This sets the stage for the next step, where we will simplify the expression by canceling out common factors. The multiplication process is a fundamental operation in algebra, and understanding it thoroughly is essential for manipulating algebraic expressions effectively. It allows us to combine different terms and expressions into a single, more manageable form. Let's proceed to the next step to see how we can further simplify this expression.

2. Simplifying the Numerator and Denominator

Multiply the coefficients and combine the variables:

4x4×22y326y5×4x3=88x4y3104x3y5\frac{4 x^4 \times 22 y^3}{26 y^5 \times 4 x^3} = \frac{88 x^4 y^3}{104 x^3 y^5}

In this step, we perform the multiplication in both the numerator and the denominator. We multiply the numerical coefficients (4 and 22 in the numerator, 26 and 4 in the denominator) and combine the variables with their respective exponents. This results in a single fraction with a simplified numerator and denominator. The key to this step is to accurately multiply the coefficients and correctly apply the rules of exponents when combining variables. This process lays the groundwork for the final simplification, where we will cancel out common factors between the numerator and denominator. Understanding this step is crucial for efficiently manipulating algebraic expressions and preparing them for further analysis or computation.

3. Reducing the Fraction

Now, we reduce the fraction by canceling out common factors. First, simplify the numerical coefficients:

88104=8×118×13=1113\frac{88}{104} = \frac{8 \times 11}{8 \times 13} = \frac{11}{13}

Next, simplify the variables using the quotient rule for exponents ($\frac{xa}{xb} = x^{a-b}$):

x4x3=x4−3=x1=x\frac{x^4}{x^3} = x^{4-3} = x^1 = x

y3y5=y3−5=y−2=1y2\frac{y^3}{y^5} = y^{3-5} = y^{-2} = \frac{1}{y^2}

Combining these results, we get:

88x4y3104x3y5=1113×x×1y2=11x13y2\frac{88 x^4 y^3}{104 x^3 y^5} = \frac{11}{13} \times x \times \frac{1}{y^2} = \frac{11x}{13y^2}

This step is the heart of the simplification process. We begin by reducing the numerical fraction to its simplest form by canceling out the greatest common factor (GCF) of the numerator and denominator. Then, we apply the quotient rule of exponents to simplify the variable terms. This rule allows us to divide terms with the same base by subtracting their exponents. The result is a simplified expression with no common factors between the numerator and denominator. This step demonstrates the power of mathematical rules and techniques in making complex expressions more manageable and understandable. The ability to reduce fractions and simplify variable terms is essential for solving algebraic problems and expressing results in their most concise form.

Final Answer

Therefore, the simplified form of the given expression is:

11x13y2\frac{11 x}{13 y^2}

This is the final result, achieved through a systematic process of multiplication and simplification. We started with a complex expression and, by applying the rules of algebra, transformed it into a more concise and manageable form. This final answer represents the original expression in its simplest terms, making it easier to work with in further calculations or analyses. The journey from the initial expression to this final simplified form highlights the importance of understanding and applying mathematical principles. It also underscores the beauty of mathematics in its ability to reduce complexity and reveal the underlying elegance of mathematical relationships. This result not only provides the answer to the problem but also reinforces the valuable skills of algebraic manipulation and simplification.

Conclusion

In summary, simplifying algebraic expressions involves a series of steps, including multiplying fractions, applying exponent rules, and reducing common factors. The process demonstrated here provides a clear methodology for tackling such problems. Mastering these techniques is crucial for success in algebra and beyond. The ability to simplify expressions not only makes mathematical problems easier to solve but also enhances one's understanding of the underlying mathematical concepts. This skill is a cornerstone of mathematical proficiency and is essential for tackling more advanced topics in mathematics and related fields. By practicing these techniques and understanding the principles behind them, you can build a strong foundation in algebra and confidently approach complex mathematical challenges. The journey of simplification is not just about finding the answer; it's about developing a deeper appreciation for the elegance and power of mathematics.