Hey guys! Today, we're diving deep into the world of exponential equations and tackling a specific problem that will help you understand the core concepts involved. Exponential equations, at first glance, might seem intimidating, but with a systematic approach and a solid grasp of exponent rules, they become quite manageable. So, let's jump right in and learn how to solve them like pros!
The Exponential Equation at Hand
Our mission, should we choose to accept it, is to solve the following exponential equation for x:
2^(6x - 8) * 64^(2 - x) = 2^(9x + 4)
Don't worry if it looks like a jumbled mess of numbers and exponents right now. We'll break it down step by step, making it crystal clear. The key to solving exponential equations lies in manipulating the equation so that we have the same base on both sides. Once we achieve that, we can simply equate the exponents and solve for our variable, x. Let's get started!
Step 1 Standardize the bases
In this initial crucial step of solving exponential equations, our primary goal is to standardize the bases. This means we want to express all terms in the equation using the same base. Looking at our equation:
2^(6x - 8) * 64^(2 - x) = 2^(9x + 4)
We can see that we already have a term with a base of 2, which is excellent. However, we also have a term with a base of 64. This is where our knowledge of exponents comes into play. We need to recognize that 64 can be expressed as a power of 2. Specifically, 64 is equal to 2 raised to the power of 6 (2^6). This realization is the cornerstone of simplifying the equation. By expressing 64 as 2^6, we bring all terms under the common base of 2, paving the way for further simplification and ultimately solving for x.
So, let's rewrite 64 as 2^6 in our equation:
2^(6x - 8) * (2^6)^(2 - x) = 2^(9x + 4)
Now, the equation looks more uniform, with all terms expressed in terms of base 2. This standardization is a critical step because it allows us to apply the properties of exponents effectively. In the next step, we'll leverage these properties to further simplify the equation and bring it closer to a solvable form.
Step 2 Leverage the Power of a Power Rule
Now that we've standardized the base, it's time to put the power of a power rule into action! This rule states that when you raise a power to another power, you multiply the exponents. In mathematical terms, (am)n = a^(m*n). This rule is a fundamental tool in simplifying exponential expressions, and it's exactly what we need to tackle the term (26)(2 - x) in our equation.
Applying the power of a power rule to our equation:
2^(6x - 8) * (2^6)^(2 - x) = 2^(9x + 4)
We multiply the exponents 6 and (2 - x), which gives us 6 * (2 - x) = 12 - 6x. So, the term (26)(2 - x) simplifies to 2^(12 - 6x). Replacing this in our equation, we get:
2^(6x - 8) * 2^(12 - 6x) = 2^(9x + 4)
Notice how the equation is becoming cleaner and more manageable with each step. We've successfully eliminated the parentheses and combined the exponents within the second term. This simplification brings us closer to our goal of having a single exponential term on each side of the equation. In the next step, we'll utilize another key property of exponents to combine the terms on the left-hand side, further streamlining our equation-solving process.
Step 3 Product of Powers Rule
Having tamed the power of a power, we now turn our attention to the product of powers rule. This rule is another gem in our exponent-manipulation toolkit. It states that when you multiply powers with the same base, you add the exponents. Mathematically, this is expressed as a^m * a^n = a^(m+n). This rule is precisely what we need to combine the two exponential terms on the left-hand side of our equation:
2^(6x - 8) * 2^(12 - 6x) = 2^(9x + 4)
We have two terms, both with the base 2, being multiplied. According to the product of powers rule, we should add their exponents. So, we add (6x - 8) and (12 - 6x):
(6x - 8) + (12 - 6x) = 6x - 8 + 12 - 6x
Notice something beautiful here: the 6x and -6x terms cancel each other out! This leaves us with -8 + 12, which simplifies to 4. Therefore, the left-hand side of our equation becomes 2^4.
Our equation now looks much simpler:
2^4 = 2^(9x + 4)
We've successfully combined the exponential terms on the left-hand side into a single term. This simplification is a significant step forward because it allows us to directly compare the exponents on both sides of the equation. In the next step, we'll do just that, setting the stage for solving for our unknown variable, x.
Step 4 Equate the Exponents
We've arrived at a pivotal point in our equation-solving journey. Our equation now stands as a testament to the power of exponent manipulation:
2^4 = 2^(9x + 4)
The beauty of this form lies in its simplicity. We have the same base (2) on both sides of the equation. This means that for the equation to hold true, the exponents must be equal. This is the fundamental principle that allows us to equate the exponents and transition from an exponential equation to a linear equation.
Therefore, we can confidently state:
4 = 9x + 4
We've successfully transformed our exponential equation into a linear equation! This is a major breakthrough because linear equations are much easier to solve. We've effectively bypassed the complexities of exponents and reduced the problem to a familiar algebraic form. The next step involves isolating x and finding its value. We're on the home stretch now!
Step 5 Solve the Linear Equation
Having successfully equated the exponents, we're now faced with a straightforward linear equation:
4 = 9x + 4
Solving for x in this equation is a classic algebraic exercise. Our goal is to isolate x on one side of the equation. To do this, we'll first subtract 4 from both sides of the equation. This maintains the balance of the equation while moving us closer to isolating the term with x:
4 - 4 = 9x + 4 - 4
This simplifies to:
0 = 9x
Now, we have 9x equal to 0. To isolate x, we need to divide both sides of the equation by 9:
0 / 9 = 9x / 9
This gives us:
0 = x
Therefore, the solution to our linear equation is x = 0. But remember, we started with an exponential equation. So, the question remains: is this the solution to our original problem? The next step is crucial: we must verify our solution to ensure it satisfies the initial exponential equation.
Step 6 Verify the Solution
We've arrived at a potential solution, x = 0, but before we declare victory, it's crucial to verify that this value actually satisfies our original exponential equation:
2^(6x - 8) * 64^(2 - x) = 2^(9x + 4)
To verify, we'll substitute x = 0 back into the original equation and see if both sides are equal. Let's start by plugging in x = 0:
2^(6(0) - 8) * 64^(2 - 0) = 2^(9(0) + 4)
Now, let's simplify each side of the equation:
Left-hand side:
2^(6(0) - 8) * 64^(2 - 0) = 2^(-8) * 64^2
Remember that 64 is 2^6, so we can rewrite 64^2 as (26)2, which is 2^12 (using the power of a power rule):
2^(-8) * 2^12
Now, using the product of powers rule, we add the exponents: -8 + 12 = 4. So, the left-hand side simplifies to 2^4.
Right-hand side:
2^(9(0) + 4) = 2^(0 + 4) = 2^4
We see that both the left-hand side and the right-hand side simplify to 2^4. This confirms that our solution, x = 0, is indeed a valid solution to the original exponential equation. We've successfully navigated the complexities of the equation and arrived at the correct answer!
The Final Answer
After meticulously working through each step, from standardizing the base to verifying our solution, we've confidently arrived at the answer. The solution to the exponential equation:
2^(6x - 8) * 64^(2 - x) = 2^(9x + 4)
is:
x = 0
So there you have it, guys! We've conquered this exponential equation together. Remember, the key to success lies in understanding the properties of exponents and applying them systematically. Keep practicing, and you'll become an exponential equation-solving master in no time!
Key Takeaways for Exponential Equation Mastery
Before we wrap up, let's quickly recap the key takeaways from our equation-solving adventure. These are the core principles and techniques that will empower you to tackle a wide range of exponential equations with confidence:
- Standardize the Bases: This is often the first and most crucial step. Look for ways to express all terms in the equation using the same base. This unlocks the power of exponent rules and simplifies the equation significantly.
- Power of a Power Rule: Remember that (am)n = a^(m*n). This rule allows you to simplify expressions where a power is raised to another power. It's a workhorse in exponential equation manipulation.
- Product of Powers Rule: Don't forget that a^m * a^n = a^(m+n). When multiplying powers with the same base, add the exponents. This rule is essential for combining exponential terms.
- Equate the Exponents: Once you have the same base on both sides of the equation, you can equate the exponents. This transforms the exponential equation into a simpler algebraic equation.
- Solve the Linear Equation: After equating the exponents, you'll typically be left with a linear equation. Use basic algebraic techniques to isolate the variable and find its value.
- Verify the Solution: Always, always, always verify your solution by plugging it back into the original equation. This ensures that your solution is valid and prevents extraneous solutions.
By mastering these key takeaways, you'll be well-equipped to tackle any exponential equation that comes your way. Keep practicing, stay curious, and remember that even the most complex equations can be solved with a systematic approach and a solid understanding of the fundamentals.
Practice Problems to Sharpen Your Skills
Now that we've dissected the solution and highlighted the key concepts, it's time to put your newfound knowledge to the test! The best way to master exponential equations is through practice. So, here are a few practice problems to help you sharpen your skills:
- Solve for x: 3^(2x + 1) = 81
- Solve for x: 5^(x - 2) = 125
- Solve for x: 4^(3x) * 16^(x - 1) = 4^7
- Solve for x: 2^(x + 3) = 8^(x - 1)
- Solve for x: 9^(2x) = 27^(x + 1)
Try solving these problems on your own, applying the techniques and principles we discussed. Remember to standardize the bases, use the power rules, equate the exponents, and verify your solutions. If you get stuck, revisit the steps we outlined in the main solution. With consistent practice, you'll develop a strong intuition for solving exponential equations.
Don't be afraid to experiment and explore different approaches. The more you practice, the more comfortable and confident you'll become. Solving exponential equations is a valuable skill in mathematics and various scientific fields. So, embrace the challenge, enjoy the process, and watch your problem-solving abilities soar!
