Simplifying Expressions: (20x^5y^2 / 5x^{-3}y^7)^{-3}

Have you ever encountered a complex algebraic expression and felt overwhelmed? You're not alone! Simplifying expressions is a fundamental skill in mathematics, and it's crucial for solving equations, understanding mathematical concepts, and even for practical applications in various fields. In this article, we'll break down the process of simplifying the expression $\left(\frac{20 x^5 y^2}{5 x^{-3} y^7}\right)^{-3}$, where $x \neq 0$ and $y \neq 0$. We'll take a step-by-step approach, explaining the rules and techniques involved, so you can confidently tackle similar problems in the future.

Understanding the Expression

Before diving into the simplification process, let's take a closer look at the expression we're dealing with: $\left(\frac{20 x^5 y^2}{5 x^{-3} y^7}\right)^{-3}$.

This expression involves several components:

• Variables: We have two variables, x and y, which represent unknown quantities.
• Coefficients: The numbers 20 and 5 are coefficients, which are constants that multiply the variables.
• Exponents: The variables have exponents, such as 5, 2, -3, and 7, which indicate the power to which the variable is raised. A negative exponent means we're dealing with a reciprocal.
• Fraction: The expression inside the parentheses is a fraction, representing a division operation.
• Outer Exponent: The entire fraction is raised to the power of -3.

Our goal is to simplify this expression by applying the rules of exponents and algebraic manipulations. We'll aim to reduce the expression to its simplest form, where the variables have the lowest possible exponents and the coefficients are simplified.

Step-by-Step Simplification

Let's break down the simplification process into manageable steps:

Step 1: Simplify Inside the Parentheses

Our first step is to simplify the expression inside the parentheses: $\frac{20 x^5 y^2}{5 x^{-3} y^7}$.

We can start by simplifying the coefficients. Divide 20 by 5:

$\frac{20}{5} = 4$

Now, let's deal with the variables. We'll use the rule of exponents that states: $\frac{a^m}{a^n} = a^{m-n}$.

For the x terms, we have $\frac{x^5}{x^{-3}}$. Applying the rule, we get:

$x^{5 - (-3)} = x^{5 + 3} = x^8$

For the y terms, we have $\frac{y^2}{y^7}$. Applying the rule, we get:

$y^{2 - 7} = y^{-5}$

So, after simplifying the fraction inside the parentheses, we have:

$4x^8y^{-5}$

Step 2: Apply the Outer Exponent

Now we need to apply the outer exponent of -3 to the simplified expression: $(4x^8y^{-5})^{-3}$.

We'll use the rule of exponents that states: $(a^m)^n = a^{m \cdot n}$. This means we need to multiply each exponent inside the parentheses by -3.

First, consider the coefficient 4. We can rewrite 4 as $4^1$, so we have:

$4^{1 \cdot (-3)} = 4^{-3}$

Next, for the x term, we have:

$x^{8 \cdot (-3)} = x^{-24}$

For the y term, we have:

$y^{-5 \cdot (-3)} = y^{15}$

So, after applying the outer exponent, we get:

$4^{-3}x^{-24}y^{15}$

Step 3: Eliminate Negative Exponents

To further simplify the expression, we need to eliminate the negative exponents. We'll use the rule of exponents that states: $a^{-n} = \frac{1}{a^n}$.

For the term $4^{-3}$, we have:

$4^{-3} = \frac{1}{4^3} = \frac{1}{64}$

For the term $x^{-24}$, we have:

$x^{-24} = \frac{1}{x^{24}}$

Now, substitute these back into our expression:

$\frac{1}{64} \cdot \frac{1}{x^{24}} \cdot y^{15}$

Step 4: Combine the Terms

Finally, we can combine the terms to get our simplified expression:

$\frac{y^{15}}{64x^{24}}$

Summary of Steps

To recap, here are the steps we followed to simplify the expression:

1. Simplify Inside the Parentheses: We simplified the fraction inside the parentheses by dividing the coefficients and applying the rule for dividing exponents.
2. Apply the Outer Exponent: We applied the outer exponent of -3 to each term inside the parentheses by multiplying the exponents.
3. Eliminate Negative Exponents: We rewrote terms with negative exponents as their reciprocals with positive exponents.
4. Combine the Terms: We combined the terms to get the final simplified expression.

Key Rules and Concepts

Throughout this simplification process, we used several key rules and concepts of exponents. Let's summarize them:

• Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$ (when dividing powers with the same base, subtract the exponents).
• Power of a Power: $(a^m)^n = a^{m \cdot n}$ (when raising a power to another power, multiply the exponents).
• Negative Exponent: $a^{-n} = \frac{1}{a^n}$ (a negative exponent indicates the reciprocal of the base raised to the positive exponent).

Understanding and applying these rules is crucial for simplifying algebraic expressions effectively.

Common Mistakes to Avoid

When simplifying expressions, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

• Incorrectly Applying the Quotient of Powers Rule: Remember to subtract the exponents when dividing powers with the same base. A common mistake is to divide the exponents instead.
• Forgetting to Distribute the Outer Exponent: When raising a product or quotient to a power, remember to apply the exponent to every term inside the parentheses, including coefficients.
• Misunderstanding Negative Exponents: A negative exponent indicates a reciprocal, not a negative number. For example, $x^{-2}$ is equal to $\frac{1}{x^2}$, not $-x^2$.
• Arithmetic Errors: Simple arithmetic errors can derail the entire simplification process. Double-check your calculations, especially when dealing with negative numbers and fractions.

By being aware of these common mistakes, you can increase your accuracy and confidence in simplifying expressions.

Practice Problems

To solidify your understanding, let's work through a few more practice problems. Try simplifying these expressions on your own, and then check your answers against the solutions provided.

Problem 1: Simplify $\left(\frac{9 a^4 b^{-2}}{3 a^{-1} b^5}\right)^2$

Solution:

1. Simplify inside the parentheses: $\frac{9 a^4 b^{-2}}{3 a^{-1} b^5} = 3a^{4-(-1)}b^{-2-5} = 3a^5b^{-7}$
2. Apply the outer exponent: $(3a^5b^{-7})^2 = 3^2a^{5 \cdot 2}b^{-7 \cdot 2} = 9a^{10}b^{-14}$
3. Eliminate negative exponents: $9a^{10}b^{-14} = \frac{9a^{10}}{b^{14}}$

Problem 2: Simplify $\left(\frac{16 x^{-3} y^6}{4 x^2 y^{-1}}\right)^{-1}$

Solution:

1. Simplify inside the parentheses: $\frac{16 x^{-3} y^6}{4 x^2 y^{-1}} = 4x^{-3-2}y^{6-(-1)} = 4x^{-5}y^7$
2. Apply the outer exponent: $(4x^{-5}y^7)^{-1} = 4^{-1}x^{-5 \cdot (-1)}y^{7 \cdot (-1)} = 4^{-1}x^5y^{-7}$
3. Eliminate negative exponents: $4^{-1}x^5y^{-7} = \frac{x^5}{4y^7}$

By working through these practice problems, you can reinforce your understanding of the simplification process and build your problem-solving skills.

Conclusion

Simplifying algebraic expressions is a fundamental skill in mathematics. By understanding the rules of exponents and applying them systematically, you can confidently tackle complex expressions and reduce them to their simplest forms. In this article, we've walked through the process of simplifying the expression $\left(\frac{20 x^5 y^2}{5 x^{-3} y^7}\right)^{-3}$, highlighting the key steps and concepts involved. Remember to practice regularly and pay attention to common mistakes to avoid. With consistent effort, you'll master the art of simplifying expressions and unlock new levels of mathematical understanding.

For more in-depth information and resources on simplifying expressions and other mathematical concepts, consider exploring reputable websites like Khan Academy. They offer a wealth of tutorials, practice exercises, and explanations to help you excel in mathematics.