Show That N = M + 5 If (81^(m+1) + 3(9^(2m)))/(3^(m-2) × 9^(m-1)) = 28 × 3^n

Introduction

In this article, we delve into an intriguing algebraic problem involving exponents and simplification. Our main task is to demonstrate that if the equation (81^(m+1) + 3(9^(2m)))/(3(m-2) × 9^(m-1)) = 28 × 3^n holds true, then the relationship n = m + 5 must also be valid. This requires a meticulous step-by-step approach, leveraging the properties of exponents and algebraic manipulation to transform the initial equation into a form that clearly reveals the connection between n and m. This exploration is crucial for anyone looking to enhance their skills in algebraic problem-solving, particularly in the context of exponential equations. Understanding how to simplify complex expressions and derive relationships between variables is a fundamental aspect of mathematics, applicable in various fields such as engineering, physics, and computer science. Let’s embark on this mathematical journey to uncover the elegance and precision embedded in exponential equations.

Understanding the Problem

Before diving into the solution, it's essential to grasp the core of the problem. We are given an equation that combines exponential terms with different bases, primarily 3 and 9, and a constant. The equation is structured as a fraction equal to another exponential term multiplied by a constant. Our mission is to show that a specific relationship, n = m + 5, exists between the exponents n and m. This task requires us to simplify the given expression, isolate the exponential terms, and eventually compare the exponents on both sides of the equation. The challenge lies in effectively applying the rules of exponents to simplify the complex expression and reveal the inherent relationship between n and m. To achieve this, we will use properties such as the power of a power rule, the product of powers rule, and the quotient of powers rule. These rules are the fundamental tools that allow us to manipulate exponential expressions and simplify them into a more manageable form. This initial understanding sets the stage for a systematic and logical approach to solving the problem.

Simplifying the Left-Hand Side (LHS)

The left-hand side (LHS) of the equation is (81^(m+1) + 3(9^(2m)))/(3(m-2) × 9^(m-1)). Our first step is to express all terms with a common base, which in this case is 3. We know that 81 = 3^4 and 9 = 3^2. Substituting these values, we get:

((3⁴⁾(m+1) + 3((3²⁾(2m)))/(3^(m-2) × (3²⁾(m-1))

Now, we apply the power of a power rule, which states that (a^b)c = a^(bc). This simplifies the expression to:

(3^(4(m+1)) + 3(3^(4m)))/(3(m-2) × 3^(2(m-1)))

Expanding the exponents, we have:

(3^(4m+4) + 3^(4m+1))/(3(m-2) × 3^(2m-2))

Next, we apply the product of powers rule, which states that a^b × a^c = a^(b+c), to the denominator:

(3^(4m+4) + 3^(4m+1))/(3((m-2)+(2m-2)))

This simplifies the denominator to:

(3^(4m+4) + 3^(4m+1))/(3(3m-4))

Now, we can factor out the common factor 3^(4m+1) from the numerator:

(3^(4m+1)(33 + 1))/(3^(3m-4))

This gives us:

(3^(4m+1)(27 + 1))/(3^(3m-4))

Which simplifies to:

(3^{(4m+1)(28))/(3}(3m-4))

Finally, we apply the quotient of powers rule, which states that a^b / a^c = a^(b-c):

28 × 3^((4m+1)-(3m-4))

Simplifying the exponent, we get:

28 × 3^(4m + 1 - 3m + 4)

Which results in:

28 × 3^(m+5)

Thus, the simplified LHS is 28 × 3^(m+5).

Equating LHS and RHS

Now that we have simplified the left-hand side (LHS) of the equation, we can equate it to the right-hand side (RHS) and solve for the relationship between n and m. The simplified LHS is 28 × 3^(m+5), and the given RHS is 28 × 3^n. Equating these two, we have:

28 × 3^(m+5) = 28 × 3^n

Since the coefficients (28) are the same on both sides, we can divide both sides by 28, which gives us:

3^(m+5) = 3^n

Now, we have an equation where the bases are the same. According to the property of exponential equations, if a^b = a^c, then b = c. Applying this to our equation, we can equate the exponents:

m + 5 = n

This clearly shows that n = m + 5, which is the relationship we were asked to prove. This step is crucial as it directly links the simplified forms of both sides of the original equation, allowing us to derive the desired relationship. The process of equating the LHS and RHS and then comparing the exponents is a fundamental technique in solving exponential equations. It highlights the power of simplification and the importance of understanding the properties of exponents. By reducing the equation to its simplest form, we can easily identify the connection between the variables and arrive at the solution.

Conclusion: Proving n = m + 5

In conclusion, we have successfully demonstrated that if (81^(m+1) + 3(9^(2m)))/(3(m-2) × 9^(m-1)) = 28 × 3^n, then n = m + 5. This was achieved through a series of methodical steps, starting with the simplification of the left-hand side of the equation. We converted all terms to a common base of 3, applied the power of a power rule, the product of powers rule, and the quotient of powers rule to reduce the expression to its simplest form. Factoring out common terms and simplifying exponents allowed us to transform the complex fraction into 28 × 3^(m+5). Equating this simplified form with the right-hand side, 28 × 3^n, led us to the direct relationship n = m + 5 by equating the exponents. This exercise underscores the importance of understanding and applying the properties of exponents in solving algebraic problems. The systematic approach we employed, from initial simplification to final deduction, highlights a valuable problem-solving strategy applicable to a wide range of mathematical challenges. The ability to manipulate exponential expressions and derive relationships between variables is a key skill in mathematics and its applications.

This problem not only reinforces the fundamental principles of algebra but also demonstrates the elegance and interconnectedness of mathematical concepts. The journey from the initial complex equation to the final, simple relationship illustrates the power of mathematical reasoning and the beauty of its solutions.