Solving for Y in Exponential Equations: A Comprehensive Guide

Understanding and solving equations involving exponents is a fundamental aspect of advanced mathematics. In this article, we will delve into a specific problem, 33x 3y-2 3-2, where the goal is to determine the value of y. This problem not only requires basic knowledge of exponents but also illustrates the process of equating the exponents when the bases are the same.

Problem Statement

The given equation is:

33x 3y-2 3-2

Step-by-Step Solution

Step 1: Combine the Exponents on the Left-Hand Side (LHS)

First, we need to combine the exponents on the left-hand side using the properties of exponents:

33x 3y-2 33 (y-2) 33y-2

Thus, the equation simplifies to:

33y-2 3-2

Step 2: Equate the Exponents

Since the bases are the same, we can equate the exponents:

3y-2 -2

Step 3: Solve for Y

Now, we solve the equation for y:

3y-2 -2

Add 2 to both sides of the equation:

3y 0

Divide both sides by 3:

y 0/3 -3

Therefore, the value of y is -3.

Further Examples and Exercises

Let's explore a few more examples to ensure a deeper understanding of the concept.

Example 1

Given the equation:

3y1 3-2

Since the bases are the same, we equate the exponents:

y1 -2

Thus, y1 -3.

Example 2

Given the equation:

33y-2 1/32

We can rewrite 1/32 as:

3-2 33y-2

Equating the exponents:

3y-2 -2

Solving for y:

3y 0

y 0/3 -3

Example 3

Given the equation:

34y-2 1/32

First, rewrite 1/32 as:

3-2 34y-2

Equating the exponents:

4y-2 -2

Solving for y:

4y 0

y 0/4 0

Conclusion

Solving for y in exponential equations involves a few key steps: combining exponents, equating them, and solving the resulting equation. Understanding and practicing these steps can greatly enhance your ability to handle similar problems in advanced mathematics.