Finding The Value Of N In The Equation (2x^9y^n)(4x^2y^10) = 8x^11y^20

In the realm of algebra, solving equations is a fundamental skill. This article delves into the process of finding the value of 'n' that satisfies the given equation: (2x^9yn)(4x^2y10) = 8x^11y20. We will break down the equation, apply the rules of exponents, and systematically isolate 'n' to arrive at the solution. This problem not only tests your understanding of algebraic manipulation but also reinforces the core principles of exponents and polynomial multiplication.

Deciphering the Equation: A Step-by-Step Approach

To begin, let's rewrite the equation to clearly see each component:

(2x^9yn)(4x^2y10) = 8x^11y20

Our primary goal here is to determine the unknown value of 'n'. To achieve this, we need to understand the fundamental rules of multiplying expressions with exponents. When multiplying terms with the same base, we add their exponents. For example, x^a * x^b = x^(a+b). This principle will be crucial in simplifying our equation. Now, we proceed by multiplying the coefficients (the numerical parts) and the variables separately. Multiplying the coefficients, we have 2 * 4 = 8, which already matches the coefficient on the right side of the equation. This indicates that our focus should now shift to the variables and their respective exponents.

For the 'x' terms, we have x^9 * x^2. Applying the rule of exponents, we add the exponents: 9 + 2 = 11. This gives us x^11, which perfectly matches the x^11 term on the right side of the equation. This confirms that our calculations are on the right track and that the 'x' components of the equation are balanced. The remaining task is to address the 'y' terms. We have y^n * y^10 on the left side and y^20 on the right side. Again, applying the rule of exponents, we add the exponents of 'y': n + 10. This sum must be equal to the exponent of 'y' on the right side, which is 20. Thus, we have formed a simple equation: n + 10 = 20. Solving this equation for 'n' will give us the value that satisfies the original equation. By isolating 'n', we can uncover the solution to this algebraic puzzle.

Isolating 'n': Unveiling the Solution

Having simplified the equation to n + 10 = 20, we can now isolate 'n' to find its value. This involves performing basic algebraic manipulation. Our goal is to get 'n' by itself on one side of the equation. To do this, we need to eliminate the '+ 10' from the left side. The inverse operation of addition is subtraction, so we will subtract 10 from both sides of the equation. This maintains the balance of the equation, ensuring that the equality remains valid.

Subtracting 10 from both sides, we get:

n + 10 - 10 = 20 - 10

This simplifies to:

n = 10

Therefore, the value of 'n' that makes the equation true is 10. This solution means that when we substitute 10 for 'n' in the original equation, the left side will indeed equal the right side. We can verify this by plugging 10 back into the original equation and checking if the equality holds true. This step-by-step process demonstrates the power of algebraic manipulation in solving equations and finding unknown variables. By applying the rules of exponents and performing basic operations, we have successfully determined the value of 'n' that satisfies the given equation.

Verification: Ensuring the Accuracy of Our Solution

To ensure the accuracy of our solution, we can substitute the value of n = 10 back into the original equation:

(2x^9yn)(4x^2y10) = 8x^11y20

Replacing 'n' with 10, we get:

(2x^9y10)(4x^2y10) = 8x^11y20

Now, let's multiply the terms on the left side. First, multiply the coefficients: 2 * 4 = 8. Next, multiply the 'x' terms: x^9 * x^2 = x^(9+2) = x^11. Finally, multiply the 'y' terms: y^10 * y^10 = y^(10+10) = y^20. Combining these results, we have:

8x^11y20 = 8x^11y20

This confirms that our solution is correct, as the left side of the equation now exactly matches the right side. This verification step is crucial in mathematics, as it provides assurance that the answer we have obtained is indeed the correct one. By substituting the value back into the original equation, we have rigorously tested our solution and demonstrated its validity. This process not only reinforces our understanding of the problem but also highlights the importance of accuracy and verification in mathematical problem-solving.

Conclusion: The Value of 'n' Unveiled

In conclusion, by carefully applying the rules of exponents and algebraic manipulation, we have successfully determined that the value of n that makes the equation (2x^9yn)(4x^2y10) = 8x^11y20 true is 10. This problem serves as a valuable exercise in understanding the fundamental principles of algebra and the importance of systematic problem-solving. The ability to manipulate equations, isolate variables, and verify solutions is crucial in mathematics and various other fields. By mastering these skills, one can confidently tackle more complex algebraic challenges and apply mathematical reasoning to real-world problems. This exercise underscores the significance of a solid foundation in algebra and its role in developing critical thinking and problem-solving abilities. The journey to finding 'n' has not only provided us with a solution but has also reinforced our understanding of the elegant and powerful language of mathematics.

What is the value of n that satisfies the equation (2x^9yn)(4x^2y10) = 8x^11y20?

Finding the Value of n in the Equation (2x^9yn)(4x^2y10) = 8x^11y20