Solve For { P $} : : : { 4^{2p} = 8^{-2p-1} \}

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will focus on solving the equation $4^{2p} = 8^{-2p-1}$, which involves manipulating exponential expressions and applying logarithmic properties. By the end of this discussion, you will have a clear understanding of how to approach and solve similar exponential equations.

Understanding Exponential Equations

Exponential equations involve variables in the exponent, and they can be written in the form $a^x = b$, where $a$ is the base, $x$ is the exponent, and $b$ is the result. In the given equation, $4^{2p}$ and $8^{-2p-1}$ are both exponential expressions, and we need to find the value of $p$ that makes them equal.

Step 1: Simplify the Equation

To simplify the equation, we can start by expressing both sides with the same base. Since $4$ and $8$ are both powers of $2$, we can rewrite the equation as follows:

$$\left(2^2\right)^{2p} = \left(2^3\right)^{-2p-1}$$

Using the property of exponents that $(a^m)^n = a^{mn}$, we can simplify the equation further:

$$2^{4p} = 2^{-6p-3}$$

Step 2: Equate the Exponents

Since the bases are the same, we can equate the exponents:

$$4p = -6p-3$$

Step 3: Solve for p

To solve for $p$, we can add $6p$ to both sides of the equation:

$$10p = -3$$

Then, we can divide both sides by $10$:

$$p = -\frac{3}{10}$$

Conclusion

Solving the equation $4^{2p} = 8^{-2p-1}$ required us to simplify the equation, equate the exponents, and solve for $p$. By following these steps, we were able to find the value of $p$ that makes the equation true. This process can be applied to similar exponential equations, and it is essential to understand the underlying principles of exponents and logarithms.

Tips and Tricks

• When solving exponential equations, it is essential to simplify the equation by expressing both sides with the same base.
• Equating the exponents is a crucial step in solving exponential equations.
• Solving for the variable requires careful manipulation of the equation.

Real-World Applications

Exponential equations have numerous real-world applications, including:

• Finance: Exponential equations are used to calculate compound interest and investment returns.
• Science: Exponential equations are used to model population growth, chemical reactions, and other phenomena.
• Engineering: Exponential equations are used to design and optimize systems, such as electrical circuits and mechanical systems.

Common Mistakes

• Failing to simplify the equation by expressing both sides with the same base.
• Equating the wrong exponents or variables.
• Not checking the solution for extraneous solutions.

Practice Problems

1. Solve the equation $3^{2x} = 9^{-x-1}$.
2. Solve the equation $2^{3x} = 8^{x-2}$.
3. Solve the equation $5^{x-2} = 25^{2x+1}$.

Solutions

1. $x = -\frac{1}{2}$
2. $x = \frac{1}{4}$
3. $x = -\frac{3}{2}$