Solving For X And Y In The Equation (a^3 B^4 / A^2 B)^6 = A^x B^y

#equation #exponents #algebra

This article provides a detailed solution for finding the values of $x$ and $y$ in the equation $\left(\frac{a^3 b^4}{a^2 b}\right)^6 = a^x b^y$. This problem involves simplifying expressions with exponents and understanding the rules of exponents. We will break down each step to ensure a clear and comprehensive understanding. This is a common type of problem in algebra and is crucial for mastering exponent manipulation.

Understanding the Problem

Before diving into the solution, let's make sure we fully grasp the problem. We are given an equation where an expression involving variables $a$ and $b$ is raised to the power of 6. This expression is then equated to another expression with $a$ and $b$ raised to the powers of $x$ and $y$, respectively. Our goal is to find the values of these exponents, $x$ and $y$. This involves simplifying the left-hand side of the equation using exponent rules and then comparing the exponents of $a$ and $b$ on both sides.

The equation we need to solve is:

$$\left(\frac{a^3 b^4}{a^2 b}\right)^6 = a^x b^y$$

We need to find the values for $x$ and $y$ that make this equation true.

Step-by-Step Solution

Let's proceed step-by-step to solve this equation. We will begin by simplifying the expression inside the parentheses and then apply the power of 6.

Step 1: Simplify the Expression Inside the Parentheses

We have the expression $\frac{a^3 b^4}{a^2 b}$. To simplify this, we will use the quotient rule of exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$. Applying this rule to both $a$ and $b$ terms, we get:

$$\frac{a^3 b^4}{a^2 b} = a^{3-2} b^{4-1} = a^1 b^3 = ab^3$$

So, the expression inside the parentheses simplifies to $ab^3$.

Step 2: Apply the Power of 6

Now, we need to raise the simplified expression $ab^3$ to the power of 6. Using the power of a product rule, which states that $(ab)^n = a^n b^n$, and the power of a power rule, which states that $(a^m)^n = a^{mn}$, we can proceed as follows:

$$(ab^3)^6 = a^6 (b^3)^6 = a^6 b^{3 \times 6} = a^6 b^{18}$$

Thus, the left-hand side of the equation becomes $a^6 b^{18}$.

Step 3: Equate the Exponents

Now our equation looks like this:

$$a^6 b^{18} = a^x b^y$$

For this equation to hold true, the exponents of $a$ and $b$ on both sides must be equal. Therefore, we can equate the exponents:

• For $a$: $x = 6$
• For $b$: $y = 18$

Hence, we have found the values of $x$ and $y$.

Final Answer

The values of $x$ and $y$ that satisfy the equation are:

• $x = 6$
• $y = 18$

Therefore, the solution to the equation $\left(\frac{a^3 b^4}{a^2 b}\right)^6 = a^x b^y$ is $x = 6$ and $y = 18$.

Key Concepts Used

This problem primarily uses the following exponent rules:

1. Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$
2. Power of a Product Rule: $(ab)^n = a^n b^n$
3. Power of a Power Rule: $(a^m)^n = a^{mn}$

These rules are fundamental in simplifying expressions involving exponents and are crucial in solving algebraic problems. Understanding and applying these rules correctly is essential for success in algebra and beyond. Practice with various problems can help solidify these concepts.

Additional Practice Problems

To further enhance your understanding, here are a few practice problems similar to the one we just solved:

1. $\left(\frac{x^5 y^2}{x^2 y}\right)^3 = x^a y^b$. Find $a$ and $b$.
2. $\left(\frac{p^4 q^6}{p q^3}\right)^2 = p^m q^n$. Find $m$ and $n$.
3. $\left(\frac{c^7 d^3}{c^4 d}\right)^4 = c^u d^v$. Find $u$ and $v$.

Working through these problems will provide you with more practice in applying the exponent rules and help you become more confident in solving such equations.

Common Mistakes to Avoid

When working with exponents, it's easy to make mistakes if you're not careful. Here are some common mistakes to avoid:

1. Incorrectly Applying the Quotient Rule: Ensure you subtract the exponents in the correct order. It's $a^{m-n}$, not $a^{n-m}$ when dividing $a^m$ by $a^n$.
2. Forgetting the Power of a Product Rule: When raising a product to a power, remember to apply the power to each factor in the product. For example, $(ab)^n = a^n b^n$, not just $a^n$ or $b^n$.
3. Misapplying the Power of a Power Rule: When raising a power to another power, multiply the exponents. So, $(a^m)^n = a^{mn}$, not $a^{m+n}$.
4. Not Simplifying Inside Parentheses First: It's generally a good practice to simplify the expression inside parentheses before applying any exponents outside the parentheses. This can make the problem easier to manage.

By being aware of these common pitfalls, you can avoid making these errors and improve your accuracy in solving exponent-related problems.

Real-World Applications

While manipulating exponents might seem like an abstract mathematical concept, it has numerous real-world applications. Exponents are used extensively in various fields, including:

1. Science: In physics, exponents are used to represent very large or very small numbers in scientific notation. For example, the speed of light is approximately $3 \times 10^8$ meters per second.
2. Computer Science: Exponents are fundamental in computer science, particularly in representing data sizes (e.g., kilobytes, megabytes, gigabytes) and computational complexity.
3. Finance: Compound interest calculations involve exponents. The formula for compound interest, $A = P(1 + r/n)^{nt}$, uses exponents to calculate the future value of an investment.
4. Engineering: Exponents are used in various engineering calculations, such as determining the power of a signal or the growth rate of a population.
5. Mathematics: Exponents are the foundation for many mathematical concepts, including logarithms, exponential functions, and polynomial expressions.

Understanding exponents is therefore not just a matter of mastering algebraic techniques but also of preparing oneself for a wide range of practical applications in various fields.

Conclusion

In this article, we have thoroughly explored the solution to the equation $\left(\frac{a^3 b^4}{a^2 b}\right)^6 = a^x b^y$. We found that $x = 6$ and $y = 18$ by systematically applying the rules of exponents. Mastering these rules is crucial for simplifying algebraic expressions and solving equations. By understanding the quotient rule, the power of a product rule, and the power of a power rule, you can confidently tackle similar problems.

Remember, practice is key to mastering any mathematical concept. Work through additional problems, and don't hesitate to review the fundamental rules whenever needed. With consistent effort, you will become proficient in manipulating exponents and solving algebraic equations. This skill is not only valuable in academic settings but also has wide-ranging applications in various fields, from science and engineering to finance and computer science. Keep practicing, and you'll find that exponents become a powerful tool in your mathematical toolkit.

We also highlighted some common mistakes to avoid when working with exponents, such as misapplying the quotient rule or forgetting the power of a product rule. By being mindful of these potential pitfalls, you can minimize errors and improve your accuracy. Furthermore, we discussed the real-world applications of exponents, illustrating their relevance in various fields and emphasizing the importance of understanding this mathematical concept.

By continuing to explore and practice with exponents, you'll strengthen your mathematical foundation and enhance your ability to solve complex problems. Whether you're a student learning algebra or a professional applying mathematical principles in your field, a solid understanding of exponents will serve you well.