Simplify (x^7 Y^6) / (x Y^4): A Step-by-Step Guide

Dec 5, 2025 by Alex Johnson 51 views

In this article, we'll break down the process of simplifying the algebraic expression $\frac{x^7 y^6}{x y^4}$ . This involves using the basic rules of exponents to combine like terms and reduce the expression to its simplest form. Let’s dive in!

Understanding the Basics of Exponents

Before we jump into the simplification, it's important to understand the fundamental rules of exponents. Exponents, also known as powers, represent how many times a number (the base) is multiplied by itself. For example, $x^7$ means x is multiplied by itself seven times (x * x * x * x * x * x * x). When we divide terms with the same base, we subtract the exponents.

Rule 1: Division of Powers with the Same Base: When dividing like bases, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$ .
Rule 2: Variable Terms: In our expression, we have both x and y terms. We apply the division rule separately to each of these.
Rule 3: Simplifying: Simplifying expressions makes them easier to understand and work with. In algebra, this often means reducing an expression to its smallest possible terms.

When approaching an algebraic problem, always start by identifying the like terms. In the expression $\frac{x^7 y^6}{x y^4}$ , our like terms are the 'x' terms ( $x^7$ and $x$ ) and the 'y' terms ( $y^6$ and $y^4$ ). Treat each set of like terms separately using the rules of exponents. Remember, when a variable doesn't have an explicit exponent, it is understood to have an exponent of 1. So, 'x' is the same as $x^1$ .

Correct application of these exponent rules is critical in many areas of mathematics, including algebra, calculus, and more advanced topics. Mastering these basics provides a strong foundation for tackling complex problems. For instance, in calculus, you'll often encounter expressions that need to be simplified before differentiation or integration. Therefore, a solid grasp of simplifying expressions is indispensable.

Moreover, understanding exponents helps simplify complex computations in physics and engineering. Whether you're calculating the growth rate of populations, or understanding the behavior of algorithms in computer science, exponents are your friend.

Step-by-Step Simplification

Let's simplify the expression $\frac{x^7 y^6}{x y^4}$ step-by-step.

Step 1: Rewrite the Expression

First, rewrite the expression to clearly show each term:

$\frac{x^7 y^6}{x^1 y^4}$

This helps visualize the exponents and makes the next steps easier.

Step 2: Apply the Quotient Rule to x Terms

Now, apply the quotient rule to the x terms: $\frac{x^7}{x^1} = x^{7-1} = x^6$ .

This step involves subtracting the exponent in the denominator from the exponent in the numerator. So, $x^7$ divided by $x^1$ becomes $x^6$ .

Step 3: Apply the Quotient Rule to y Terms

Next, apply the same rule to the y terms: $\frac{y^6}{y^4} = y^{6-4} = y^2$ .

Similarly, $y^6$ divided by $y^4$ simplifies to $y^2$ .

Step 4: Combine the Simplified Terms

Finally, combine the simplified x and y terms to get the final simplified expression:

$x^6 y^2$

Therefore, $\frac{x^7 y^6}{x y^4}$ simplifies to $x^6 y^2$ .

Common Mistakes to Avoid

When simplifying expressions, it's easy to make common mistakes. Here are a few to watch out for:

Forgetting the Exponent of 1: Remember that if a variable doesn't have an explicit exponent, it's understood to be 1. For example, x is the same as $x^1$ .
Incorrectly Applying the Quotient Rule: Ensure you are subtracting the exponents correctly. It's always (exponent in numerator) - (exponent in denominator).
Mixing Up Multiplication and Division: The rule for multiplying like bases is to add the exponents, while the rule for dividing is to subtract them. Don't mix these up!
Ignoring Negative Exponents: Sometimes, you might end up with negative exponents. Remember that $x^{-n} = \frac{1}{x^n}$ .

To avoid these mistakes, always double-check your work and ensure each step follows the correct exponent rules. Practice regularly to build confidence and accuracy. Make sure you understand each step and why it is done that way.

Practice Problems

To solidify your understanding, let's try a few practice problems.

Simplify $\frac{a^5 b^3}{a^2 b}$ .
Simplify $\frac{p^9 q^4}{p^3 q^2}$ .
Simplify $\frac{m^{10} n^7}{m^5 n^3}$ .

Solutions

$\frac{a^5 b^3}{a^2 b} = a^{5-2} b^{3-1} = a^3 b^2$ .
$\frac{p^9 q^4}{p^3 q^2} = p^{9-3} q^{4-2} = p^6 q^2$ .
$\frac{m^{10} n^7}{m^5 n^3} = m^{10-5} n^{7-3} = m^5 n^4$ .

By working through these problems, you can reinforce your understanding and build your skills in simplifying algebraic expressions.

Real-World Applications

Simplifying algebraic expressions isn't just an academic exercise; it has real-world applications in various fields. Let's explore a few examples.

Physics: In physics, you often need to simplify equations to solve for unknown variables. For instance, simplifying kinematic equations or electrical circuit equations requires a solid understanding of algebraic simplification.
Engineering: Engineers use algebraic simplification to design structures, analyze systems, and optimize performance. Whether it's civil, mechanical, or electrical engineering, simplifying expressions is a fundamental skill.
Computer Science: In computer science, simplifying expressions is crucial for optimizing algorithms and writing efficient code. Simplifying Boolean expressions, for example, can lead to faster and more reliable software.
Economics: Economists use algebraic simplification to model economic systems, analyze market trends, and make predictions. Simplifying supply and demand equations, for example, helps in understanding market equilibrium.

By mastering algebraic simplification, you're not just learning a mathematical skill; you're equipping yourself with a valuable tool that can be applied across various disciplines.

Conclusion

Simplifying the expression $\frac{x^7 y^6}{x y^4}$ involves understanding and applying the basic rules of exponents. By subtracting the exponents of like terms, we reduced the expression to its simplest form: $x^6 y^2$ . Remember to avoid common mistakes and practice regularly to build your skills. With a solid understanding of these concepts, you'll be well-equipped to tackle more complex algebraic problems. Keep practicing, and you'll master these skills in no time! For more in-depth explanations and examples, visit Khan Academy's Algebra Section.