Solving The Equation √((x-2)³) = 27 A Step-by-Step Guide
Hey there, math enthusiasts! Ever stumbled upon an equation that looks like a tangled mess of roots and exponents? Well, today we're going to dive headfirst into one such equation and emerge victorious with the solution. Our quest? To solve the equation √((x-2)³) = 27 and uncover the value of 'x'. Buckle up, because we're about to embark on a mathematical adventure!
Decoding the Equation: A Step-by-Step Approach
1. Understanding the Landscape: Breaking Down the Equation
Before we start swinging our mathematical swords, let's take a moment to understand the equation we're facing. We have √((x-2)³) = 27, which involves a square root, an exponent, and a variable 'x' lurking within. To conquer this, we need to strategically undo the operations and isolate 'x'. Remember, guys, in math, it's all about peeling back the layers like an onion! We will use the properties of exponents and roots to simplify the equation. The main goal is to isolate x on one side of the equation, which will reveal its true value.
First, let's rewrite the square root as an exponent. Recall that the square root of something is the same as raising it to the power of 1/2. So, we can rewrite the equation as: ((x-2)³)^(1/2) = 27. This transformation is crucial as it allows us to apply the rules of exponents more easily. Next, we'll use the power of a power rule, which states that (am)n = a^(m*n). Applying this rule to our equation, we get: (x-2)^(3 * (1/2)) = 27, which simplifies to: (x-2)^(3/2) = 27. We're making progress, guys! We've managed to get rid of the square root and express the left side with a single exponent.
Now, let's think about how to get rid of that pesky 3/2 exponent. The key is to raise both sides of the equation to the reciprocal of 3/2, which is 2/3. Why 2/3? Because when we multiply exponents, (3/2) * (2/3) equals 1, effectively canceling out the exponent on the left side. So, we raise both sides to the power of 2/3: ((x-2)(3/2))(2/3) = 27^(2/3). Applying the power of a power rule again, the left side simplifies to (x-2)^1, which is just (x-2). On the right side, we have 27^(2/3). Let's tackle that. Remember, a fractional exponent like 2/3 means we're taking the cube root (the denominator 3) and then squaring the result (the numerator 2). So, 27^(2/3) is the same as (∛27)². The cube root of 27 is 3 (since 3 * 3 * 3 = 27), so we have 3². And 3² is simply 9. Our equation now looks much cleaner: x-2 = 9. We're in the home stretch!
2. Isolating the Unknown: Unmasking the Value of 'x'
Our equation has now transformed into a much simpler form: x - 2 = 9. The mission now is crystal clear: isolate 'x' on one side of the equation. This is a classic algebraic maneuver, guys, and it's as simple as adding 2 to both sides of the equation. Why add 2? Because adding 2 to the left side will cancel out the -2, leaving 'x' all by its lonesome. So, we add 2 to both sides: x - 2 + 2 = 9 + 2. This simplifies to: x = 11. Hallelujah! We've found it! The value of 'x' that satisfies the equation is 11.
3. The Grand Finale: Verification and Assurance
Before we declare victory and march off into the sunset, let's do a quick check to make sure our answer is correct. We'll plug x = 11 back into the original equation: √((x-2)³) = 27. Substituting x = 11, we get: √((11-2)³) = 27. This simplifies to: √((9)³) = 27. Now, 9³ is 9 * 9 * 9, which equals 729. So, we have: √729 = 27. And what's the square root of 729? It's indeed 27! So, our equation holds true. We've successfully verified our solution, and we can confidently say that x = 11 is the correct answer. High five! You can always check your solution by plugging it back into the original equation. This is a crucial step in mathematics, guys, because it ensures that your answer is correct and satisfies the original problem. By plugging x = 11 back into the original equation, we can confirm that it makes the equation true. This gives us confidence in our solution and solidifies our understanding of the problem-solving process.
Why This Matters: The Significance of Solving Equations
Solving equations like this isn't just an academic exercise, guys. It's a fundamental skill that has wide-ranging applications in various fields. From physics and engineering to economics and computer science, equations are the language of the universe. Being able to manipulate and solve them allows us to model real-world phenomena, make predictions, and design solutions to complex problems. The ability to solve equations is not just about getting the right answer; it's about developing a logical and systematic approach to problem-solving. This skill is transferable to many other areas of life, helping us to break down complex challenges into smaller, manageable steps and to find solutions through careful analysis and reasoning.
Real-World Connections: Equations in Action
Imagine you're designing a bridge, for example. You'd need to calculate the forces acting on the bridge and ensure that it can withstand those forces. This involves setting up and solving equations. Or, consider a scientist studying the spread of a disease. They might use equations to model the infection rate and predict how many people will be affected. Even in everyday situations, we use equations without realizing it. Calculating how much time it will take to drive to a certain destination involves solving an equation (distance = speed × time). So, the skills we learn in solving equations are not just theoretical; they have practical applications in the real world. Understanding the principles of equation solving empowers us to make informed decisions and to solve problems in a variety of contexts. It helps us develop critical thinking skills and a systematic approach to problem-solving that can be applied to many different areas of life.
Beyond the Basics: Exploring Advanced Equations
While this equation might seem challenging at first, it's just the tip of the iceberg. There's a whole universe of equations out there, from simple linear equations to complex differential equations. As you delve deeper into mathematics, you'll encounter equations that require more sophisticated techniques to solve. But the fundamental principles we've discussed here – understanding the operations, isolating the variable, and verifying the solution – will remain essential. So, embrace the challenge, keep practicing, and don't be afraid to tackle even the most daunting equations. With perseverance and the right tools, you can conquer any mathematical problem that comes your way. Remember, math is not just about memorizing formulas; it's about understanding the underlying concepts and developing a logical way of thinking. The more you practice solving equations, the more comfortable and confident you will become in your mathematical abilities. This confidence will not only help you in your math studies but also in many other areas of your life.
Wrapping Up: The Power of Mathematical Thinking
So, there you have it, folks! We've successfully solved the equation √((x-2)³) = 27 and discovered that x = 11. But more importantly, we've learned a valuable lesson about problem-solving. By breaking down the equation into smaller steps, applying the rules of algebra, and carefully verifying our solution, we've demonstrated the power of mathematical thinking. Keep exploring, keep questioning, and keep solving, guys. The world of mathematics is full of fascinating challenges waiting to be conquered! We've shown that by understanding the underlying principles and applying a systematic approach, we can tackle even the most complex mathematical problems. The key is to break down the problem into smaller, manageable steps and to use the tools and techniques we've learned to solve each step. And remember, guys, practice makes perfect! The more you practice solving equations, the better you will become at it.
So, the next time you encounter a challenging equation, don't be intimidated. Remember the steps we've discussed, take a deep breath, and dive in. You might be surprised at what you can achieve. And always remember, mathematics is not just a subject in school; it's a way of thinking, a way of approaching problems, and a way of understanding the world around us.
x = 11