Solving (1/16)^x = 2^(3x+7): A Step-by-Step Guide

by Alex Johnson 50 views

Are you struggling with exponential equations? Do you find yourself scratching your head when faced with problems like (1/16)^x = 2^(3x+7)? Fear not! This comprehensive guide will walk you through the process step-by-step, breaking down each concept and technique so you can confidently tackle similar problems in the future. We'll explore the fundamental principles of exponents, demonstrate how to manipulate equations, and ultimately arrive at the solution. So, let's dive in and conquer this mathematical challenge together!

Understanding Exponential Equations

Before we jump into solving the equation, let's first establish a solid understanding of what exponential equations are and the key principles that govern them. An exponential equation is an equation in which the variable appears in the exponent. These types of equations often model real-world phenomena like population growth, radioactive decay, and compound interest. Solving them requires a good grasp of exponent rules and algebraic manipulation.

The core idea behind solving exponential equations is to express both sides of the equation with the same base. This allows us to equate the exponents and solve for the variable. Remember these crucial exponent rules:

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

These rules will be our arsenal as we navigate through the solution process. We will be using these rules to rewrite our equation in a simplified form. Understanding these rules and how to apply them is the cornerstone of solving exponential equations. Make sure you are comfortable with them before moving on to the next section.

Step 1: Expressing Both Sides with the Same Base

Now, let's tackle our specific equation: (1/16)^x = 2^(3x+7). The first critical step is to express both sides of the equation using the same base. Notice that 16 is a power of 2 (16 = 2^4). This is our key to simplifying the equation. We can rewrite 1/16 as 2^(-4). Why? Because, as we learned earlier, a^(-m) = 1/a^(m). So, 1/16 = 1/2^4 = 2^(-4).

Substituting this back into our original equation, we get:

(2(-4))x = 2^(3x+7)

Now, we can apply another exponent rule: (a(m))(n) = a^(m*n). This simplifies the left side of the equation:

2^(-4x) = 2^(3x+7)

See how we've transformed the equation? Both sides now have the same base (2). This is a significant milestone because it allows us to move to the next step: equating the exponents. By expressing both sides with the same base, we've created a foundation for solving the equation. This step is crucial for any exponential equation, so always look for ways to express numbers as powers of a common base.

Step 2: Equating the Exponents

With both sides of the equation expressed with the same base (2), we can now equate the exponents. This is a fundamental principle in solving exponential equations: if a^(m) = a^(n), then m = n. Applying this principle to our equation, 2^(-4x) = 2^(3x+7), we can directly equate the exponents:

-4x = 3x + 7

We've successfully transformed the exponential equation into a simple linear equation! This is a huge step forward. Now, we have a familiar algebraic problem to solve. Our next goal is to isolate the variable 'x'. This involves using basic algebraic manipulations like adding or subtracting terms from both sides of the equation.

Equating the exponents is the linchpin in solving exponential equations. It allows us to move from the realm of exponents to the more familiar territory of linear equations. The ability to make this transition is what makes solving these problems manageable.

Step 3: Solving for x

Now that we have the linear equation -4x = 3x + 7, we can solve for 'x'. The first step is to gather all the 'x' terms on one side of the equation. Let's add 4x to both sides:

-4x + 4x = 3x + 7 + 4x

This simplifies to:

0 = 7x + 7

Next, we want to isolate the 'x' term. Subtract 7 from both sides:

0 - 7 = 7x + 7 - 7

This simplifies to:

-7 = 7x

Finally, to solve for 'x', divide both sides by 7:

-7 / 7 = 7x / 7

This gives us the solution:

x = -1

We've done it! We've successfully solved for 'x'. This algebraic manipulation is a standard part of solving many types of equations. The key is to perform the same operations on both sides of the equation to maintain balance and isolate the variable.

Step 4: Verifying the Solution

It's always a good practice to verify your solution by plugging it back into the original equation. This ensures that our answer is correct and that we haven't made any errors along the way. Let's substitute x = -1 into the original equation: (1/16)^x = 2^(3x+7)

(1/16)^(-1) = 2^(3(-1)+7)*

Simplify the left side: (1/16)^(-1) = 16^(1) = 16

Simplify the right side: 2^(3*(-1)+7) = 2^(-3+7) = 2^(4) = 16

Since both sides of the equation are equal (16 = 16), our solution x = -1 is verified. This step is a crucial safeguard against errors and gives us confidence in our answer. Verification reinforces the understanding of the solution process and the correctness of the final result.

Conclusion

In this guide, we've walked through the process of solving the exponential equation (1/16)^x = 2^(3x+7) step-by-step. We started by understanding the fundamental principles of exponential equations, then we expressed both sides of the equation with the same base, equated the exponents, solved the resulting linear equation, and finally, verified our solution. By following these steps, you can confidently solve similar exponential equations.

Remember, the key to success in mathematics is practice. The more problems you solve, the better you'll become at recognizing patterns and applying the appropriate techniques. Don't be afraid to make mistakes – they are valuable learning opportunities. Keep practicing, and you'll master exponential equations in no time!

To further enhance your understanding of exponential equations and related mathematical concepts, I highly recommend exploring resources like Khan Academy's Exponential Equations Section. It offers a wealth of instructional videos, practice exercises, and articles to deepen your knowledge.