Solving (1/4)^(3z-1) = 16^(z+2) * 64^(z-2) A Step-by-Step Guide

by ADMIN 64 views
Iklan Headers

Hey guys! Today, we're diving into the fascinating world of exponential equations, specifically tackling the problem: (1/4)^(3z-1) = 16^(z+2) * 64^(z-2). Don't worry if it looks intimidating at first glance; we'll break it down step by step, making sure everyone understands the underlying concepts and how to apply them. Our goal is to not only solve this particular equation but also to equip you with the skills to confidently tackle similar problems in the future. Exponential equations are fundamental in mathematics and have wide applications in various fields, including physics, engineering, and finance. Mastering them is crucial for anyone pursuing these areas or simply looking to enhance their mathematical prowess.

Understanding the Fundamentals of Exponential Equations

Before we jump into the solution, let's quickly review the basics. An exponential equation is an equation where the variable appears in the exponent. The key to solving these equations lies in manipulating them so that both sides have the same base. Once we achieve this, we can equate the exponents and solve for the variable. This is because of a fundamental property of exponential functions: if a^m = a^n, then m = n, provided that a is a positive number not equal to 1. This property is the cornerstone of our approach to solving exponential equations. Think of it like this: if two powers with the same base are equal, then their exponents must also be equal. Understanding this principle makes solving these equations much more intuitive and less about memorizing steps.

Another important concept is the laws of exponents. These laws provide us with the tools to manipulate exponential expressions. Some of the most commonly used laws include:

  • Product of powers: a^(m) * a^(n) = a^(m+n)
  • Quotient of powers: a^(m) / a^(n) = a^(m-n)
  • Power of a power: (a(m))(n) = a^(m*n)
  • Negative exponent: a^(-n) = 1/a^(n)

These laws allow us to rewrite exponential expressions in different forms, which is crucial for simplifying equations and achieving a common base. For example, the negative exponent rule is particularly useful when dealing with fractions as bases, as we will see in our problem. The power of a power rule helps us simplify expressions where an exponent is raised to another exponent. Remember, the goal is always to manipulate the equation in a way that makes it easier to work with, and the laws of exponents are our primary tools for doing so. Mastering these laws will significantly improve your ability to solve exponential equations.

Step-by-Step Solution to (1/4)^(3z-1) = 16^(z+2) * 64^(z-2)

Now, let’s get our hands dirty and solve the given equation: (1/4)^(3z-1) = 16^(z+2) * 64^(z-2). Our first task is to express all the terms with the same base. Notice that 1/4, 16, and 64 can all be written as powers of 2. This is a common strategy in solving exponential equations: identify a common base and rewrite all terms using that base. It simplifies the equation and makes it easier to manipulate.

  1. Rewrite the bases as powers of 2:

    • 1/4 = 2^(-2)
    • 16 = 2^(4)
    • 64 = 2^(6)

    By expressing each term as a power of 2, we've set the stage for using the laws of exponents to simplify the equation. This step is crucial because it allows us to directly compare the exponents once we've simplified further.

  2. Substitute these values back into the original equation:

    (2(-2))(3z-1) = (2(4))(z+2) * (2(6))(z-2)

    Now we have an equation where all the bases are the same. This is a significant step forward because it allows us to apply the power of a power rule to further simplify the equation.

  3. Apply the power of a power rule:

    2^(-2(3z-1)) = 2^(4(z+2)) * 2^(6(z-2))

    Multiplying the exponents, we get:

    2^(-6z+2) = 2^(4z+8) * 2^(6z-12)

    The power of a power rule is essential here as it allows us to eliminate the parentheses and combine the exponents. Remember, (am)n = a^(m*n). This step is a direct application of this rule.

  4. Apply the product of powers rule on the right side:

    2^(-6z+2) = 2^(4z+8 + 6z-12)

    Simplifying the exponent on the right side:

    2^(-6z+2) = 2^(10z-4)

    The product of powers rule (a^m * a^n = a^(m+n)) is used here to combine the two exponential terms on the right side into a single term. This further simplifies the equation and brings us closer to our goal of equating the exponents.

  5. Equate the exponents:

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

    -6z + 2 = 10z - 4

    This is the critical step where we leverage the fundamental property of exponential functions: if a^m = a^n, then m = n. By equating the exponents, we transform the exponential equation into a simple linear equation.

  6. Solve for z:

    Add 6z to both sides:

    2 = 16z - 4

    Add 4 to both sides:

    6 = 16z

    Divide by 16:

    z = 6/16

    Simplify the fraction:

    z = 3/8

    And there you have it! We've successfully solved for z. This final step involves basic algebraic manipulation to isolate z. Remember to simplify your answer whenever possible.

Verifying the Solution

It's always a good practice to verify our solution by plugging it back into the original equation. This helps ensure that we haven't made any mistakes along the way. Let's plug z = 3/8 back into the original equation:

(1/4)^(3(3/8)-1) = 16^(3/8+2) * 64^(3/8-2)

Let's simplify both sides:

  • Left Side: (1/4)^(9/8-1) = (1/4)^(1/8) = (2(-2))(1/8) = 2^(-1/4)

  • Right Side: 16^(3/8+2) * 64^(3/8-2) = 16^(19/8) * 64^(-13/8) = (24)(19/8) * (26)(-13/8) = 2^(19/2) * 2^(-39/4) = 2^(38/4) * 2^(-39/4) = 2^(-1/4)

Since both sides are equal, our solution z = 3/8 is correct! Verifying your solution is a crucial step in problem-solving. It provides confidence in your answer and helps catch any potential errors.

Common Mistakes to Avoid

When solving exponential equations, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them.

  • Incorrectly applying the laws of exponents: Make sure you understand and correctly apply the product of powers, quotient of powers, and power of a power rules. A common mistake is adding exponents when they should be multiplied or vice versa. Always double-check which rule applies in each situation.
  • Forgetting to distribute: When dealing with expressions like a^(n(m+p)), remember to distribute the 'n' to both 'm' and 'p'. For example, (24)(z+2) becomes 2^(4z+8), not 2^(4z+2). Failing to distribute can lead to significant errors in your solution.
  • Not finding a common base: The key to solving exponential equations is to express all terms with the same base. If you skip this step or try to equate exponents without a common base, you'll likely arrive at an incorrect answer. Spend time identifying the common base and rewriting each term accordingly.
  • Algebraic errors: Simple algebraic mistakes can derail your solution. Be careful when adding, subtracting, multiplying, and dividing, especially when dealing with negative numbers and fractions. It's always a good idea to double-check your algebra to ensure accuracy.

By being mindful of these common mistakes, you can increase your accuracy and confidence in solving exponential equations.

Practice Problems

To solidify your understanding, try solving these practice problems:

  1. 5^(2x-1) = 125
  2. 9^(x+2) = 27^(2x-1)
  3. (1/2)^(x+3) = 4^(2x)

Working through these problems will give you hands-on experience and help you internalize the steps involved in solving exponential equations. Remember to focus on finding a common base, applying the laws of exponents correctly, and solving the resulting equation carefully. The more you practice, the more comfortable and confident you'll become.

Conclusion

Solving exponential equations like (1/4)^(3z-1) = 16^(z+2) * 64^(z-2) might seem challenging at first, but by breaking it down into manageable steps, we can conquer them! Remember the key principles: find a common base, apply the laws of exponents, equate the exponents, and solve the resulting equation. And always, always verify your solution! Understanding these steps and practicing regularly will empower you to solve a wide range of exponential equations with confidence. Keep practicing, and you'll become an exponential equation master in no time! Happy solving!