Solving For N In The Equation 6x^n * 4x^2 = 24x^6

In the realm of algebra, equations often present themselves as puzzles, inviting us to decipher the hidden values of variables. One such intriguing equation is 6xⁿ * 4x² = 24x⁶. Our mission is to determine the value of the exponent 'n' that makes this statement true. This equation elegantly combines numerical coefficients and variables raised to powers, offering a glimpse into the fundamental principles of algebraic manipulation. To embark on this mathematical journey, we will carefully dissect the equation, applying the rules of exponents and arithmetic to isolate 'n' and reveal its true identity. So, let's delve into the heart of this equation and unravel the mystery surrounding the value of 'n'.

Dissecting the Equation: A Step-by-Step Approach

To solve for 'n' in the equation 6xⁿ * 4x² = 24x⁶, we'll embark on a step-by-step journey, carefully applying the rules of exponents and arithmetic. Our primary goal is to isolate 'n' on one side of the equation, thereby revealing its value. Let's begin by simplifying the left-hand side of the equation.

Step 1: Multiplying Coefficients

The left-hand side of the equation, 6xⁿ * 4x², presents us with two terms that need to be multiplied. The first step is to multiply the numerical coefficients, which are 6 and 4. Multiplying these coefficients together, we get 6 * 4 = 24. This simplifies the left-hand side to 24xⁿ * x².

Step 2: Applying the Product of Powers Rule

Now, we encounter the product of two exponential terms with the same base, 'x'. This is where the product of powers rule comes into play. This rule states that when multiplying exponential terms with the same base, we add their exponents. In our case, we have xⁿ and x². Applying the product of powers rule, we get xⁿ * x² = xⁿ⁺². This further simplifies the left-hand side to 24xⁿ⁺².

Step 3: Equating Exponents

With the left-hand side simplified to 24xⁿ⁺², our equation now stands as 24xⁿ⁺² = 24x⁶. Notice that both sides of the equation have the same numerical coefficient, 24. This allows us to focus on the exponential terms, xⁿ⁺² and x⁶. For the equation to hold true, the exponents of these terms must be equal. Therefore, we can equate the exponents: n + 2 = 6.

Step 4: Solving for 'n'

We've arrived at a simple algebraic equation that we can easily solve for 'n'. To isolate 'n', we need to subtract 2 from both sides of the equation. This gives us n = 6 - 2, which simplifies to n = 4. Therefore, the value of 'n' that makes the statement true is 4.

Verifying the Solution: A Crucial Step

In mathematics, verifying our solutions is a crucial step to ensure accuracy and understanding. To verify our solution, we'll substitute the value of 'n' we found (n = 4) back into the original equation and see if it holds true. The original equation is 6xⁿ * 4x² = 24x⁶. Substituting n = 4, we get 6x⁴ * 4x² = 24x⁶.

Left-Hand Side Simplification

Let's simplify the left-hand side of the equation. First, we multiply the coefficients: 6 * 4 = 24. Then, we apply the product of powers rule to the exponential terms: x⁴ * x² = x⁴⁺² = x⁶. This simplifies the left-hand side to 24x⁶.

Comparison

Now, we compare the simplified left-hand side with the right-hand side of the equation. We have 24x⁶ = 24x⁶, which is clearly a true statement. This confirms that our solution, n = 4, is indeed correct. The equation holds true when we substitute n = 4, solidifying our understanding of the problem and its solution.

The Significance of Exponents: A Deeper Dive

Exponents, the small superscript numbers that accompany variables, hold a profound significance in the world of mathematics. They provide a concise way to represent repeated multiplication, allowing us to express complex relationships in a compact and elegant manner. In the equation 6xⁿ * 4x² = 24x⁶, the exponents 'n' and 2 play a crucial role in determining the overall behavior of the equation. Understanding the rules of exponents is paramount to solving such equations and unraveling their hidden meanings.

The Power of Exponents

Exponents represent the number of times a base is multiplied by itself. For example, x² means x * x, and x⁶ means x * x * x * x * x * x. The exponent indicates the power to which the base is raised. In our equation, 'n' represents the power to which 'x' is raised in the first term, and 2 represents the power to which 'x' is raised in the second term. The exponent 6 on the right-hand side represents the final power of 'x' after the multiplication is performed.

Rules of Exponents: Guiding Principles

The rules of exponents are a set of mathematical principles that govern how exponents behave in various operations. These rules provide a systematic way to simplify expressions and solve equations involving exponents. One of the key rules we used in solving our equation is the product of powers rule, which states that when multiplying exponential terms with the same base, we add their exponents. This rule allowed us to combine xⁿ and x² into xⁿ⁺², simplifying the equation and bringing us closer to the solution.

Exponents in Real-World Applications

Exponents are not confined to the realm of pure mathematics; they find widespread applications in various real-world scenarios. From calculating compound interest to modeling population growth, exponents play a vital role in understanding and quantifying phenomena that exhibit exponential behavior. In physics, exponents are used to describe the intensity of sound and light. In computer science, exponents are fundamental to understanding the growth of algorithms and the storage capacity of digital devices. The versatility of exponents makes them an indispensable tool in various fields of study and practical applications.

Conclusion: Unveiling the Value of 'n'

In conclusion, the value of 'n' that makes the statement 6xⁿ * 4x² = 24x⁶ true is 4. We arrived at this solution by carefully dissecting the equation, applying the rules of exponents and arithmetic, and systematically isolating 'n'. The journey involved multiplying coefficients, applying the product of powers rule, equating exponents, and solving a simple algebraic equation. To ensure the accuracy of our solution, we verified it by substituting n = 4 back into the original equation and confirming that it holds true. This exercise not only revealed the value of 'n' but also underscored the importance of understanding and applying the rules of exponents in algebraic manipulations. Exponents, with their ability to represent repeated multiplication concisely, play a fundamental role in mathematics and its applications across various disciplines.