Equivalent Equation For 16^(2p) = 32^(p+3)

Introduction

In the realm of mathematics, solving exponential equations often requires a blend of algebraic manipulation and a keen understanding of exponent rules. In this comprehensive guide, we will delve into the equation 16^(2p) = 32^(p+3), dissecting the steps to identify its equivalent form. Our exploration will not only reveal the correct answer but also illuminate the underlying principles that govern exponential equations. Whether you're a student grappling with exponents or a math enthusiast seeking to refine your skills, this article will serve as your compass in the world of exponential equations.

Deconstructing the Problem

Before we dive into the solution, let's first understand the essence of the problem. We are presented with an exponential equation where the bases are different (16 and 32). To find an equivalent equation, our primary goal is to express both sides of the equation with a common base. This crucial step allows us to equate the exponents and solve for the unknown variable, p. By breaking down the problem into manageable steps, we pave the way for a clear and concise solution. This approach not only helps in solving this particular equation but also equips us with a strategy applicable to a wide range of exponential problems.

Finding the Common Base

The heart of solving this exponential equation lies in identifying a common base for both 16 and 32. The key to unlocking this lies in recognizing that both numbers can be expressed as powers of 2. Let's express 16 and 32 in terms of the base 2. 16 can be written as 2^4, and 32 can be written as 2^5. By expressing the numbers with a common base, we simplify the equation and bring it closer to a solvable form. This step is not just a mathematical manipulation; it's a strategic move that allows us to apply the properties of exponents effectively and efficiently. Mastering this skill is crucial for anyone venturing into the world of exponential equations.

Rewriting the Equation with the Common Base

Now that we've identified 2 as the common base, we can rewrite the original equation, 16^(2p) = 32^(p+3), using powers of 2. Substituting 16 with 2^4 and 32 with 2^5, we get (2⁴⁾(2p) = (2⁵⁾(p+3). This transformation is a pivotal moment in our solution journey. It's where the equation starts to take a more manageable form, setting the stage for the application of exponent rules. By rewriting the equation, we've not just changed its appearance; we've unlocked its potential for simplification and solution. This step highlights the power of strategic substitution in solving mathematical problems.

Applying the Power of a Power Rule

The next step involves wielding a fundamental rule of exponents: the power of a power rule. This rule states that (a^m)n = a^(m*n). Applying this rule to our equation, (2⁴⁾(2p) = (2⁵⁾(p+3), we multiply the exponents. On the left side, we have 2^(4 * 2p) = 2^(8p). On the right side, we have 2^(5 * (p+3)) = 2^(5p+15). The equation now transforms into 2^(8p) = 2^(5p+15). This is a significant milestone in our solution process. We've successfully simplified the equation by applying a key exponent rule, bringing us closer to isolating the variable, p. This step underscores the importance of mastering exponent rules in tackling exponential equations.

Equating the Exponents

With the bases now the same, we arrive at a crucial juncture: equating the exponents. If a^m = a^n, then m = n. Applying this principle to our equation, 2^(8p) = 2^(5p+15), we can equate the exponents, resulting in the equation 8p = 5p + 15. This step is a direct consequence of having a common base, and it's what allows us to transition from an exponential equation to a linear equation, which is much easier to solve. Equating exponents is a powerful technique that simplifies the problem significantly, making the solution more accessible. It's a testament to the elegance and efficiency of mathematical principles.

The Equivalent Equation

Therefore, the equation 2^(8p) = 2^(5p+15) is the equivalent equation to 16^(2p) = 32^(p+3). This corresponds to option C in the given choices. This is our destination, the equivalent equation we set out to find. It's not just an answer; it's the culmination of our step-by-step journey through the world of exponents. The equivalent equation encapsulates the essence of the original problem in a simplified form, ready for further analysis or application. This result underscores the importance of understanding mathematical equivalence, a concept that lies at the heart of problem-solving in mathematics.

Solving for p (Optional)

Although the question asks for the equivalent equation, let's take it a step further and solve for p. Starting with 8p = 5p + 15, we can isolate p by subtracting 5p from both sides, giving us 3p = 15. Dividing both sides by 3, we find p = 5. This exercise demonstrates how finding the equivalent equation is often a stepping stone to solving for the variable. It highlights the interconnectedness of mathematical concepts and the power of algebraic manipulation. Solving for p not only provides a numerical answer but also reinforces our understanding of the equation's behavior and characteristics.

Conclusion

In conclusion, by strategically applying the principles of exponents and algebraic manipulation, we successfully identified the equivalent equation to 16^(2p) = 32^(p+3) as 2^(8p) = 2^(5p+15). This journey through exponential equations has reinforced the importance of identifying common bases, applying exponent rules, and equating exponents. Armed with these skills, you're well-equipped to tackle a wide array of exponential problems. Mathematics, at its core, is about problem-solving, and the ability to navigate exponential equations is a valuable asset in any mathematical endeavor.

Find the equivalent equation for the exponential equation 16^(2p) = 32^(p+3).

Solving Exponential Equations Find the Equivalent of 16^(2p) = 32^(p+3)