Solving Exponential Equations Step By Step Guide
In mathematics, exponential equations are equations in which the variable appears in the exponent. Solving these equations often involves manipulating the equation to have the same base on both sides, then equating the exponents. This article provides a detailed, step-by-step guide on how to solve two such exponential equations, ensuring a clear understanding of the underlying principles and techniques.
Equation 1: 32^(y-5) = 2^(2y)
The first exponential equation we will tackle is 32^(y-5) = 2^(2y). Our primary goal here is to find the value of the variable 'y' that satisfies this equation. The key to solving exponential equations like this is to express both sides of the equation with the same base. Let's break down the solution step-by-step.
Step 1: Express Both Sides with the Same Base
The initial step in solving this exponential equation is to recognize that 32 can be expressed as a power of 2. Specifically, 32 is 2 raised to the power of 5 (32 = 2^5). By rewriting 32 as 2^5, we can express both sides of the equation with the same base, which is 2. This is a critical step because it allows us to equate the exponents later on. Rewriting the equation, we get:
(25)(y-5) = 2^(2y)
This transformation is crucial because it sets the stage for simplifying the equation further. When we have the same base on both sides, we can use the properties of exponents to simplify and solve for the variable. Understanding the relationship between numbers and their powers is fundamental in solving exponential equations, and this step highlights the importance of recognizing such relationships.
Step 2: Apply the Power of a Power Rule
Now that we have expressed 32 as 2^5, we can apply the power of a power rule, which states that (am)n = a^(m*n). This rule is essential for simplifying expressions where an exponent is raised to another exponent. In our equation, we have (25)(y-5), which can be simplified using this rule. Applying the power of a power rule to the left side of the equation, we multiply the exponents:
2^(5(y-5)) = 2^(2y)
This step is a direct application of the power of a power rule and is a cornerstone in simplifying exponential expressions. By multiplying the exponents, we reduce the complexity of the equation, making it easier to solve. It's important to remember this rule as it frequently appears in various mathematical contexts, especially when dealing with exponents and powers.
Step 3: Simplify the Exponent
Before we can equate the exponents, we need to simplify the exponent on the left side of the equation. We have 5(y-5) as the exponent, which can be simplified by distributing the 5 across the terms inside the parentheses. Distributing the 5, we get:
5 * y - 5 * 5 = 5y - 25
Thus, the left side of the equation becomes 2^(5y-25). Now our equation looks like this:
2^(5y-25) = 2^(2y)
Simplifying the exponent in this manner is a standard algebraic technique that prepares the equation for the next step, where we will equate the exponents. This simplification ensures that we are dealing with the exponents in their simplest form, making the subsequent steps more straightforward and less prone to errors.
Step 4: Equate the Exponents
The equation now has the same base (2) on both sides: 2^(5y-25) = 2^(2y). At this point, we can equate the exponents. This is a crucial step in solving exponential equations because if a^m = a^n, then m = n. Equating the exponents, we get:
5y - 25 = 2y
This equation is a simple linear equation, which is much easier to solve compared to the original exponential equation. By equating the exponents, we have effectively transformed the problem into a more manageable form. This step highlights the power of manipulating exponential equations to simpler forms by leveraging the properties of exponents.
Step 5: Solve for y
Now we have a linear equation: 5y - 25 = 2y. To solve for y, we need to isolate y on one side of the equation. First, let's subtract 2y from both sides of the equation:
5y - 2y - 25 = 2y - 2y
This simplifies to:
3y - 25 = 0
Next, we add 25 to both sides:
3y - 25 + 25 = 0 + 25
This gives us:
3y = 25
Finally, we divide both sides by 3 to solve for y:
y = 25 / 3
Thus, the solution for the first equation is y = 25 / 3. This is a fractional solution, which is perfectly acceptable as the problem statement allows for integer or fractional answers. This step-by-step solution demonstrates the process of isolating the variable and solving a linear equation, a fundamental skill in algebra.
Equation 2: 216^(7y+5) = 36^(-4y+11)
Now, let's move on to the second exponential equation: 216^(7y+5) = 36^(-4y+11). Similar to the first equation, our goal is to find the value of 'y' that satisfies this equation. We will follow a similar approach, focusing on expressing both sides of the equation with the same base.
Step 1: Express Both Sides with the Same Base
The initial step in solving this equation is to recognize that both 216 and 36 can be expressed as powers of the same base. Both numbers are powers of 6. Specifically, 216 is 6 cubed (216 = 6^3), and 36 is 6 squared (36 = 6^2). Expressing both sides with the base 6, we can rewrite the equation as:
(63)(7y+5) = (62)(-4y+11)
This transformation is a critical step in simplifying the equation. By identifying the common base, we set the stage for applying the properties of exponents and solving for the variable. Recognizing these relationships between numbers and their powers is a crucial skill in dealing with exponential equations.
Step 2: Apply the Power of a Power Rule
Now that both sides of the equation are expressed with the base 6, we can apply the power of a power rule, which states that (am)n = a^(m*n). This rule is essential for simplifying expressions where an exponent is raised to another exponent. Applying this rule to both sides of the equation, we multiply the exponents:
6^(3(7y+5)) = 6^(2(-4y+11))
This step is a straightforward application of the power of a power rule and is vital for reducing the complexity of the equation. By multiplying the exponents, we make the equation easier to manipulate and solve. The power of a power rule is a fundamental concept in algebra and is frequently used in simplifying exponential expressions.
Step 3: Simplify the Exponents
Before equating the exponents, we need to simplify the exponents on both sides of the equation. On the left side, we have 3(7y+5), and on the right side, we have 2(-4y+11). We can simplify these expressions by distributing the constants across the terms inside the parentheses. Distributing the constants, we get:
Left side: 3 * 7y + 3 * 5 = 21y + 15
Right side: 2 * -4y + 2 * 11 = -8y + 22
Thus, the equation now looks like this:
6^(21y+15) = 6^(-8y+22)
Simplifying the exponents in this manner is a standard algebraic technique that prepares the equation for the next step, where we will equate the exponents. This simplification ensures that we are dealing with the exponents in their simplest form, making the subsequent steps more straightforward and less prone to errors.
Step 4: Equate the Exponents
The equation now has the same base (6) on both sides: 6^(21y+15) = 6^(-8y+22). At this point, we can equate the exponents. This is a crucial step in solving exponential equations because if a^m = a^n, then m = n. Equating the exponents, we get:
21y + 15 = -8y + 22
This equation is a linear equation, which is much easier to solve compared to the original exponential equation. By equating the exponents, we have effectively transformed the problem into a more manageable form. This step highlights the power of manipulating exponential equations to simpler forms by leveraging the properties of exponents.
Step 5: Solve for y
Now we have a linear equation: 21y + 15 = -8y + 22. To solve for y, we need to isolate y on one side of the equation. First, let's add 8y to both sides of the equation:
21y + 8y + 15 = -8y + 8y + 22
This simplifies to:
29y + 15 = 22
Next, we subtract 15 from both sides:
29y + 15 - 15 = 22 - 15
This gives us:
29y = 7
Finally, we divide both sides by 29 to solve for y:
y = 7 / 29
Thus, the solution for the second equation is y = 7 / 29. This is a fractional solution, which is perfectly acceptable as the problem statement allows for integer or fractional answers. This step-by-step solution demonstrates the process of isolating the variable and solving a linear equation, a fundamental skill in algebra.
Conclusion
Solving exponential equations involves expressing both sides with the same base, applying the power of a power rule, simplifying exponents, equating the exponents, and then solving the resulting linear equation. By following these steps, we can efficiently solve a variety of exponential equations. In the examples provided, we found that the solution for the first equation, 32^(y-5) = 2^(2y), is y = 25 / 3, and the solution for the second equation, 216^(7y+5) = 36^(-4y+11), is y = 7 / 29. These step-by-step solutions provide a clear and concise method for tackling exponential equations and highlight the importance of understanding and applying the fundamental rules of exponents.
