Solve 2(x-4)^(3/2)=54: A Step-by-Step Guide

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the fascinating world of mathematics, specifically tackling a pretty cool equation: 2(x-4)^{rac{3}{2}}=54. If you're a math whiz or just looking to brush up on your algebra skills, you're in the right place. We're going to break down this problem, explore the steps to find the solution, and discuss why understanding these types of equations is super important in various fields. So, grab your calculators, settle in, and let's get started on unraveling this mathematical mystery together!

Understanding the Equation: What Are We Dealing With?

Alright, let's kick things off by really understanding the equation we're trying to solve: 2(x-4)^{rac{3}{2}}=54. The main goal here is to find the value of 'x' that makes this equation true. It might look a little intimidating with that fractional exponent (rac{3}{2}), but don't sweat it! We'll break it down piece by piece. This type of equation is called a radical equation because it involves a root, in this case, a square root combined with a cube. Specifically, the term (x-4)^{rac{3}{2}} can be rewritten as $(\sqrt{x-4})^3$ or $\sqrt{(x-4)^3}$. Both forms are equivalent, but sometimes one is easier to work with than the other depending on the problem. The coefficient '2' multiplying the bracket means we'll need to isolate the radical term first before we can deal with the exponent. The constant '54' on the right side is what our entire expression needs to equal. Our mission, should we choose to accept it, is to isolate 'x'. This process often involves reversing the operations that have been applied to 'x'. We'll be using inverse operations, like division to undo multiplication, and exponentiation to undo roots.

Why This Matters: Real-World Applications of Radical Equations

Now, you might be asking yourself, "Why should I care about solving equations like this?" That's a fair question, and the answer is pretty awesome. While you might not be solving 2(x-4)^{rac{3}{2}}=54 on a daily basis, the principles behind solving radical and exponential equations are fundamental in tons of real-world scenarios. Think about physics, for example. Many laws of motion, particularly those involving oscillations or wave phenomena, are described using equations with exponents and roots. In engineering, especially when designing structures or analyzing fluid dynamics, you'll encounter formulas that require solving these types of equations to predict behavior or ensure safety. Even in finance, models for compound interest or risk assessment can involve exponential functions that sometimes need to be inverted, which is essentially what we're doing here. Understanding how to manipulate these equations gives you the power to analyze, predict, and create in fields ranging from computer graphics and game development (think about curves and animations) to biology (modeling population growth or decay). So, mastering these algebraic skills isn't just about passing tests; it's about equipping yourself with tools that are genuinely useful in shaping the world around you. It's like learning a secret code that unlocks deeper understanding in many scientific and technical disciplines. Plus, the logical thinking involved in solving them sharpens your problem-solving skills in all aspects of life, not just math class. Pretty neat, right?

Step-by-Step Solution: Cracking the Code

Let's get down to business and solve 2(x-4)^{rac{3}{2}}=54. The first thing we need to do is isolate the term with the exponent. It's currently being multiplied by 2, so we'll divide both sides of the equation by 2:

$\frac{2(x-4)^{\frac{3}{2}}}{2} = \frac{54}{2}$

This simplifies to:

$(x-4)^{\frac{3}{2}} = 27$

Now, we have the term with the exponent isolated. The next step is to get rid of that $\frac{3}{2}$ exponent. To do this, we need to raise both sides of the equation to the reciprocal power, which is $\frac{2}{3}$. Remember, raising a power to another power means multiplying the exponents, and $(\frac{3}{2}) \times (\frac{2}{3}) = 1$, which will leave us with just $(x-4)^1$, or simply $(x-4)$.

$((x-4)^{\frac{3}{2}})^{\frac{2}{3}} = 27^{\frac{2}{3}}$

So, we have:

$x-4 = 27^{\frac{2}{3}}$

Now, let's evaluate $27^{\frac{2}{3}}$. This means we need to find the cube root of 27 and then square the result. The cube root of 27 is 3 (since $3 \times 3 \times 3 = 27$). Then, we square that result:

$3^2 = 9$

So, our equation becomes:

$x-4 = 9$

We're almost there! The final step is to isolate 'x' by adding 4 to both sides of the equation:

$x - 4 + 4 = 9 + 4$

$x = 13$

And there you have it! The solution to the equation is $x = 13$. We successfully navigated the fractional exponent and isolated 'x'. It's all about systematic steps and understanding the properties of exponents!

Checking Our Work: Is x=13 Really the Solution?

It's super important in math, especially when dealing with equations that have roots or fractional exponents, to always check your solution. This is because sometimes, when you square both sides of an equation (or raise them to an even power), you can introduce extraneous solutions – solutions that work in the intermediate steps but not in the original equation. Let's plug $x=13$ back into our original equation, 2(x-4)^{rac{3}{2}}=54, to make sure it holds true.

Substitute $x=13$:

$2(13-4)^{\frac{3}{2}} = 54$

First, calculate the expression inside the parentheses:

$13 - 4 = 9$

Now the equation looks like this:

$2(9)^{\frac{3}{2}} = 54$

Next, we need to evaluate $9^{\frac{3}{2}}$. Remember, this means taking the square root of 9 and then cubing the result. The square root of 9 is 3.

$(\sqrt{9})^3 = 3^3$

Now, cube 3:

$3^3 = 3 \times 3 \times 3 = 27$

So, the left side of our equation becomes:

$2 \times 27$

And $2 \times 27 = 54$.

This matches the right side of our original equation: $54 = 54$. Boom! Our solution $x=13$ is correct. This verification step is crucial and gives us confidence in our answer. It's like double-checking your work before submitting a big project – always a good idea!

Analyzing the Options: Why Other Choices Are Incorrect

We found our solution to be $x = 13$, which corresponds to option B. But what about the other options: A (5), C (22), and D (no solution)? Let's quickly see why they don't work, just to solidify our understanding.

• Option A: x = 5 If we plug in $x=5$ into the original equation, we get: $2(5-4)^{\frac{3}{2}} = 2(1)^{\frac{3}{2}} = 2(1) = 2$. Since $2 \neq 54$, $x=5$ is not the solution.

• Option C: x = 22 Let's try $x=22$: $2(22-4)^{\frac{3}{2}} = 2(18)^{\frac{3}{2}}$. Calculating $18^{\frac{3}{2}}$ involves $\sqrt{18}^3$, which is not a simple whole number. $\sqrt{18}$ is approximately $4.24$. $(4.24)^3$ is approximately $76.2$. So, $2 \times 76.2 \approx 152.4$. Clearly, $152.4 \neq 54$. So, $x=22$ is not the solution.

• Option D: No solution We found a valid solution, $x=13$. Therefore, the statement that there is "no solution" is false. It's important to note that sometimes radical equations can have no solution if, for example, the process leads to a situation like $\sqrt{\text{negative number}}$ when working with real numbers, or if a potential solution is extraneous. However, in this case, we found a perfectly good real solution.

This quick check confirms that $x=13$ is indeed the unique solution among the given options. It's always good practice to test potential answers or rule out incorrect ones systematically.

Conclusion: Mastering Algebraic Challenges

So there you have it, folks! We've successfully tackled the equation 2(x-4)^{rac{3}{2}}=54, breaking down each step and arriving at the solution $x=13$. We learned how to isolate terms with fractional exponents, use reciprocal powers to simplify them, and, most importantly, the necessity of checking our answers. Remember, the path to mastering mathematics isn't about finding answers instantly, but about understanding the process and developing the confidence to tackle new challenges. Equations like this might seem daunting at first, but with a methodical approach and a bit of practice, you can unravel even the most complex mathematical puzzles. Keep practicing, keep questioning, and never be afraid to dive into the amazing world of numbers. Until next time on Plastik Magazine, stay curious and keep solving!