Solving $243^{-y}=\left(\frac{1}{243}\right)^{3 Y} \cdot 9^{-2 Y}$ A Step-by-Step Guide

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This article delves into the step-by-step solution of the exponential equation $243^{-y}=\left(\frac{1}{243}\right)^{3 y} \cdot 9^{-2 y}$. We will explore the fundamental concepts of exponents and algebraic manipulation required to arrive at the final answer. This comprehensive guide aims to provide a clear and concise understanding of the solution process, making it accessible to anyone with a basic understanding of algebra. Understanding how to solve exponential equations is crucial in various fields, including mathematics, physics, engineering, and finance. These equations often model real-world phenomena, such as population growth, radioactive decay, and compound interest. Therefore, mastering the techniques for solving them is an essential skill.

Understanding Exponential Equations

Before we jump into the solution, let's briefly review what exponential equations are and some key properties of exponents that will be helpful. An exponential equation is an equation in which the variable appears in the exponent. These equations are solved using the properties of exponents and logarithms. The key properties we'll use in this problem are:

1. Power of a Power: $(a^m)^n = a^{m \cdot n}$
2. Product of Powers: $a^m \cdot a^n = a^{m+n}$
3. Negative Exponent: $a^{-n} = \frac{1}{a^n}$
4. Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$

These properties allow us to manipulate exponential expressions and simplify equations. They provide the foundation for solving equations where the unknown variable is part of the exponent. Mastering these rules is essential for success in algebra and other mathematical disciplines. Let's begin with the step-by-step solution to the equation.

Step-by-Step Solution

To solve the exponential equation $243^{-y}=\left(\frac{1}{243}\right)^{3 y} \cdot 9^{-2 y}$, we need to express all terms with the same base. This is a common strategy when dealing with exponential equations, as it allows us to directly compare the exponents. Recognizing that 243 and 9 are powers of 3 is the first key step in simplifying this equation. The number 243 can be written as $3^5$, and 9 can be written as $3^2$. Using this information, we can rewrite the equation in terms of the base 3.

1. Expressing all terms with base 3

We start by expressing each term in the equation using the base 3:

• $243 = 3^5$, so $243^{-y} = (3^5)^{-y}$
• $\frac{1}{243} = \frac{1}{3^5} = 3^{-5}$, so $\left(\frac{1}{243}\right)^{3y} = (3^{-5})^{3y}$
• $9 = 3^2$, so $9^{-2y} = (3^2)^{-2y}$

Substituting these expressions back into the original equation, we get:

$(3^5)^{-y} = (3^{-5})^{3y} \cdot (3^2)^{-2y}$

This step is crucial because it sets the stage for applying the power of a power rule. By expressing all terms with the same base, we've transformed the equation into a form where we can directly manipulate the exponents. This is a common and effective strategy for solving exponential equations.

2. Applying the Power of a Power Rule

Now, we apply the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$. This rule allows us to simplify the exponents on both sides of the equation. Let's apply this rule to each term:

• $(3^5)^{-y} = 3^{5 \cdot (-y)} = 3^{-5y}$
• $(3^{-5})^{3y} = 3^{-5 \cdot 3y} = 3^{-15y}$
• $(3^2)^{-2y} = 3^{2 \cdot (-2y)} = 3^{-4y}$

Substituting these simplified expressions back into the equation, we have:

$3^{-5y} = 3^{-15y} \cdot 3^{-4y}$

This step significantly simplifies the equation by reducing the complex exponents into simpler forms. The power of a power rule is a fundamental tool in manipulating exponential expressions and is used extensively in solving exponential equations.

3. Applying the Product of Powers Rule

Next, we apply the product of powers rule, which states that $a^m \cdot a^n = a^{m+n}$. This rule allows us to combine the terms on the right side of the equation that have the same base. Applying this rule, we get:

$3^{-15y} \cdot 3^{-4y} = 3^{-15y + (-4y)} = 3^{-19y}$

Now our equation looks like this:

$3^{-5y} = 3^{-19y}$

By applying the product of powers rule, we've further simplified the equation into a form where we can directly compare the exponents. This step is essential for isolating the variable and solving for its value.

4. Equating the Exponents

Since the bases are now the same (both are 3), we can equate the exponents. This is a crucial step in solving exponential equations because it transforms the problem into a simple algebraic equation. If $a^m = a^n$, then $m = n$. Applying this principle to our equation, we get:

$-5y = -19y$

Now we have a linear equation in terms of y, which is much easier to solve. This transformation is the key to finding the value of the variable in exponential equations.

5. Solving for y

To solve the linear equation $-5y = -19y$, we need to isolate y. We can do this by adding $19y$ to both sides of the equation:

$-5y + 19y = -19y + 19y$

This simplifies to:

$14y = 0$

Now, we divide both sides by 14 to solve for y:

$\frac{14y}{14} = \frac{0}{14}$

Which gives us:

$y = 0$

Therefore, the solution to the equation is $y = 0$. This means that when y is 0, the original exponential equation holds true. This final step completes the solution process, providing a definitive answer to the problem.

Verification

To ensure our solution is correct, we should substitute $y = 0$ back into the original equation and verify that both sides are equal. This step is crucial in any mathematical problem, as it helps to catch any potential errors made during the solution process. Substituting $y = 0$ into the original equation $243^{-y}=\left(\frac{1}{243}\right)^{3 y} \cdot 9^{-2 y}$, we get:

$243^{-0}=\left(\frac{1}{243}\right)^{3 \cdot 0} \cdot 9^{-2 \cdot 0}$

Simplifying the exponents, we have:

$243^0 = \left(\frac{1}{243}\right)^0 \cdot 9^0$

Any non-zero number raised to the power of 0 is 1. Therefore:

$1 = 1 \cdot 1$

$1 = 1$

Since both sides of the equation are equal, our solution $y = 0$ is correct. This verification step provides confidence in the accuracy of the solution and confirms that all the algebraic manipulations were performed correctly.

Conclusion

In this article, we successfully solved the exponential equation $243^{-y}=\left(\frac{1}{243}\right)^{3 y} \cdot 9^{-2 y}$ by expressing all terms with the same base, applying the power of a power and product of powers rules, equating the exponents, and solving the resulting linear equation. The solution we found is $y = 0$. We also verified our solution by substituting it back into the original equation. This process demonstrates the systematic approach required to solve exponential equations and highlights the importance of understanding the properties of exponents. Solving exponential equations is a fundamental skill in mathematics, with applications in various scientific and engineering fields. By mastering these techniques, you can confidently tackle more complex problems involving exponential relationships.

By understanding the properties of exponents and using algebraic manipulation, we can solve various exponential equations. This example demonstrates a clear and concise approach to solving such problems, emphasizing the importance of expressing terms with the same base and applying the rules of exponents correctly. The ability to solve exponential equations is a valuable skill in mathematics and its applications.