Solving For X In $32^{\frac{x}{3}} \cdot 32^6=32^8$ A Comprehensive Guide

Unlocking the value of x that satisfies a given equation is a fundamental skill in mathematics. In this guide, we will delve into the process of solving for x in the equation $32^{\frac{x}{3}} \cdot 32^6 = 32^8$. This equation involves exponential terms, and understanding how to manipulate these terms is crucial for arriving at the correct solution. We will break down the steps involved, explaining the underlying principles and properties of exponents. This comprehensive approach will not only help you solve this specific problem but also equip you with the knowledge to tackle similar exponential equations with confidence. Understanding the properties of exponents is key to solving equations like this. The product of powers property states that when multiplying exponents with the same base, you add the powers. In this case, we have $32^{\frac{x}{3}} \cdot 32^6$, which can be rewritten as $32^{\frac{x}{3} + 6}$. Now, the equation becomes $32^{\frac{x}{3} + 6} = 32^8$. Since the bases are the same, we can equate the exponents: $\frac{x}{3} + 6 = 8$. To solve for x, we first subtract 6 from both sides of the equation, giving us $\frac{x}{3} = 2$. Finally, we multiply both sides by 3 to isolate x, resulting in $x = 6$. Therefore, the value of x that makes the equation true is 6. This step-by-step approach ensures clarity and accuracy in solving the equation.

Step-by-Step Solution

To effectively solve for x in the equation $32^{\frac{x}{3}} \cdot 32^6 = 32^8$, we will follow a structured, step-by-step approach. This method ensures clarity and minimizes the chances of error. Each step will be explained in detail, highlighting the mathematical principles involved. By understanding the logic behind each step, you will be better equipped to solve similar problems in the future. This methodical approach is crucial for mastering exponential equations and building a strong foundation in algebra. Let's begin by outlining the steps we will take:

1. Apply the Product of Powers Property: We will use the rule that states $a^m \cdot a^n = a^{m+n}$ to simplify the left side of the equation.
2. Equate the Exponents: Once the bases are the same on both sides of the equation, we can set the exponents equal to each other.
3. Solve the Linear Equation: We will solve the resulting linear equation for x using basic algebraic techniques.
4. Verify the Solution: Finally, we will substitute the value of x back into the original equation to ensure it holds true.

1. Apply the Product of Powers Property

The initial step in solving the equation $32^{\frac{x}{3}} \cdot 32^6 = 32^8$ involves applying the product of powers property. This property is a fundamental rule of exponents that simplifies expressions involving the multiplication of powers with the same base. The rule states that when you multiply powers with the same base, you add the exponents. Mathematically, this is expressed as $a^m \cdot a^n = a^{m+n}$, where a is the base and m and n are the exponents. Applying this property allows us to combine the exponential terms on the left side of the equation into a single term, making the equation easier to manipulate and solve. This step is crucial for simplifying the equation and setting the stage for the subsequent steps in the solution process. In our case, the base is 32, and the exponents are $\frac{x}{3}$ and 6. Therefore, we can rewrite the left side of the equation as $32^{\frac{x}{3} + 6}$. This simplification is a key step in isolating x and ultimately finding its value.

$$32^{\frac{x}{3}} \cdot 32^6 = 32^{\frac{x}{3} + 6}$$

2. Equate the Exponents

After applying the product of powers property, our equation now looks like this: $32^{\frac{x}{3} + 6} = 32^8$. This step involves recognizing a crucial property of exponential equations: if the bases are the same on both sides of the equation, then the exponents must be equal. This principle allows us to transform the exponential equation into a simpler linear equation, which is much easier to solve. By equating the exponents, we eliminate the exponential terms and focus solely on the algebraic relationship between the exponents. This step is a cornerstone of solving exponential equations and is essential for isolating the variable x. The logic behind this principle is that if two powers with the same base are equal, then their exponents must also be equal. This is a direct consequence of the definition of exponents and the uniqueness of exponential functions. In our case, the base is 32 on both sides, so we can confidently equate the exponents. Therefore, we can set the exponent on the left side, $\frac{x}{3} + 6$, equal to the exponent on the right side, which is 8. This gives us the linear equation $\frac{x}{3} + 6 = 8$. This equation is now in a form that we can easily solve for x using standard algebraic techniques.

$$\frac{x}{3} + 6 = 8$$

3. Solve the Linear Equation

Now that we have the linear equation $\frac{x}{3} + 6 = 8$, the next step is to solve for x. This involves using basic algebraic operations to isolate x on one side of the equation. The first step in isolating x is to subtract 6 from both sides of the equation. This eliminates the constant term on the left side and simplifies the equation. Subtracting 6 from both sides maintains the equality and moves us closer to isolating x. After subtracting 6, we are left with the equation $\frac{x}{3} = 2$. The next step is to eliminate the fraction by multiplying both sides of the equation by 3. This isolates x on the left side and gives us the solution. Multiplying both sides by 3 maintains the equality and completes the process of solving for x. After multiplying by 3, we find that $x = 6$. This is the value of x that satisfies the linear equation and, consequently, the original exponential equation. Therefore, the solution to the equation is $x = 6$. This step demonstrates the power of algebraic manipulation in solving equations and finding unknown variables.

$$\frac{x}{3} + 6 = 8$$

$$\frac{x}{3} = 2$$

$$x = 6$$

4. Verify the Solution

To ensure the accuracy of our solution, it is crucial to verify that the value we found for x, which is 6, indeed satisfies the original equation. This step involves substituting x = 6 back into the original equation, $32^{\frac{x}{3}} \cdot 32^6 = 32^8$, and checking if both sides of the equation are equal. Verification is a critical step in problem-solving as it helps to catch any potential errors made during the solution process. It provides confidence in the correctness of the answer and reinforces the understanding of the mathematical concepts involved. If the left side of the equation equals the right side after substitution, then our solution is correct. If not, it indicates that an error was made, and we need to re-examine our steps. In this case, we will substitute x = 6 into the original equation and simplify both sides to see if they are equal. This process will confirm whether 6 is the correct value for x and validate our solution. Substituting x = 6 into the equation, we get $32^{\frac{6}{3}} \cdot 32^6 = 32^8$. Simplifying the exponent, we have $32^2 \cdot 32^6 = 32^8$. Applying the product of powers property, we get $32^{2+6} = 32^8$, which simplifies to $32^8 = 32^8$. Since both sides of the equation are equal, our solution x = 6 is verified to be correct. This final step provides assurance that we have accurately solved the equation.

$$32^{\frac{6}{3}} \cdot 32^6 = 32^8$$

$$32^2 \cdot 32^6 = 32^8$$

$$32^{2+6} = 32^8$$

$$32^8 = 32^8$$

Conclusion

In conclusion, by following a step-by-step approach and applying the properties of exponents, we have successfully solved for x in the equation $32^{\frac{x}{3}} \cdot 32^6 = 32^8$. The value of x that makes the equation true is 6. This process involved using the product of powers property to simplify the equation, equating the exponents, solving the resulting linear equation, and verifying the solution. Each step was crucial in arriving at the correct answer and understanding the underlying mathematical principles. This comprehensive guide not only provides the solution to this specific problem but also equips you with the knowledge and skills to tackle similar exponential equations. The key takeaways from this solution include the importance of understanding the properties of exponents, the methodical approach to solving equations, and the need to verify the solution to ensure accuracy. By mastering these concepts, you can confidently solve a wide range of exponential equations and further enhance your algebraic skills. This problem serves as a valuable example of how mathematical principles can be applied to solve real-world problems and reinforces the importance of a strong foundation in algebra. The ability to solve exponential equations is essential in various fields, including science, engineering, and finance. Therefore, understanding the methods and techniques discussed in this guide is highly beneficial for academic and professional pursuits. Remember to practice solving similar problems to solidify your understanding and build your problem-solving skills.

Practice Problems

To further solidify your understanding of solving exponential equations, here are some practice problems. Working through these problems will help you apply the concepts and techniques discussed in this guide and build your problem-solving skills. Each problem is designed to test your understanding of the properties of exponents and your ability to manipulate equations to solve for x. Try to solve each problem independently, following the step-by-step approach outlined earlier. After attempting the problems, you can check your answers to ensure accuracy. If you encounter any difficulties, revisit the relevant sections of this guide to refresh your understanding. Practice is key to mastering mathematical concepts, and these problems provide an excellent opportunity to reinforce your knowledge of exponential equations. By working through a variety of problems, you will develop confidence in your ability to solve them and gain a deeper understanding of the underlying principles. Remember to pay attention to the details of each problem and apply the appropriate techniques. The more you practice, the more proficient you will become at solving exponential equations. Here are a few practice problems to get you started:

1. $5^{\frac{x}{2}} \cdot 5^3 = 5^7$
2. $2^{\frac{x}{4}} \cdot 2^2 = 2^5$
3. $9^{\frac{x}{2}} \cdot 9^1 = 9^4$

Answers:

1. x = 8
2. x = 12
3. x = 6