Solving The Equation: $3(x+9)^{3/4} = 24$ - A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little math problem: solving the equation $3(x+9)^{\frac{3}{4}}=24$. Don't worry if it looks intimidating at first glance. We're going to break it down step by step, so it's super easy to follow. Solving equations like this involves understanding exponents and a bit of algebraic manipulation. So, grab your thinking caps, and let's get started! We'll go through each step meticulously to ensure you understand not just the how, but also the why behind each operation. By the end of this guide, you'll be a pro at tackling similar equations. Let's jump right in and demystify this mathematical puzzle together! Understanding these principles is crucial not only for this specific problem but also for a wide range of mathematical challenges you might encounter in the future.

Step 1: Isolate the Exponential Term

The first thing we need to do is isolate the term with the exponent, which is $(x+9)^{\frac{3}{4}}$. To do this, we'll divide both sides of the equation by 3. This is a fundamental algebraic principle: whatever operation you perform on one side of the equation, you must perform on the other to maintain balance. By isolating the exponential term, we simplify the equation and make it easier to manipulate further. This step is crucial because it sets the stage for undoing the exponent in the subsequent steps. Think of it as peeling back the layers of the equation to get to the core variable, x. It’s like preparing the canvas before you start painting – you need a clean surface to work with. So, let’s get to it and make our equation a bit more manageable!

$$3(x+9)^{\frac{3}{4}}=24$$

Divide both sides by 3:

$$(x+9)^{\frac{3}{4}} = \frac{24}{3}$$

$$(x+9)^{\frac{3}{4}} = 8$$

Step 2: Eliminate the Fractional Exponent

Now, we have $(x+9)^{\frac{3}{4}} = 8$. To get rid of the fractional exponent, we need to raise both sides of the equation to the reciprocal of the exponent, which is $\frac{4}{3}$. Remember, raising a power to its reciprocal results in an exponent of 1, effectively eliminating the exponent on the left side. This is a key technique in solving equations with fractional exponents. The reciprocal exponent acts like a magic key that unlocks the variable from the power it's trapped in. By applying this step, we're essentially reversing the exponentiation, bringing us closer to solving for x. So, let's wield this magic key and see what happens!

$$( (x+9)^{\frac{3}{4}} )^{\frac{4}{3}} = 8^{\frac{4}{3}}$$

Using the power of a power rule (which states that $(a^m)^n = a^{mn}$), the left side simplifies to:

$$(x+9)^{(\frac{3}{4} \cdot \frac{4}{3})} = 8^{\frac{4}{3}}$$

$$x+9 = 8^{\frac{4}{3}}$$

Step 3: Simplify the Right Side

Next, we need to simplify $8^{\frac{4}{3}}$. Remember that a fractional exponent can be interpreted as a combination of a root and a power. In this case, the denominator (3) represents the cube root, and the numerator (4) represents the power to which we raise the result. So, $8^{\frac{4}{3}}$ means we first take the cube root of 8, and then raise the result to the power of 4. This approach makes it easier to handle the exponent. It's like breaking down a complex task into smaller, more manageable steps. By understanding the relationship between fractional exponents, roots, and powers, we can simplify even the most daunting-looking expressions. Let's tackle this simplification and see what we get!

The cube root of 8 is 2, since $2^3 = 8$. So, we have:

$$8^{\frac{4}{3}} = (8^{\frac{1}{3}})^4 = 2^4$$

Now, we calculate $2^4$:

$$2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16$$

So, our equation now looks like this:

$$x+9 = 16$$

Step 4: Isolate x

We're almost there! Now, to isolate x, we need to subtract 9 from both sides of the equation. This is another fundamental algebraic principle: to isolate a variable, we perform the inverse operation. In this case, the inverse operation of addition is subtraction. By subtracting 9 from both sides, we're effectively moving the constant term to the right side, leaving x all alone on the left. This step brings us to the final solution, revealing the value of x that satisfies the original equation. Let's complete this final step and unveil the answer!

$$x+9-9 = 16-9$$

$$x = 7$$

Step 5: Check Your Solution

It's always a good idea to check your solution by plugging it back into the original equation. This helps ensure that you haven't made any mistakes along the way. Plugging the solution back into the original equation is like proofreading your work – it's a crucial step in ensuring accuracy. By verifying that our solution satisfies the equation, we gain confidence in our answer and demonstrate a thorough understanding of the problem-solving process. This step not only confirms our result but also reinforces the underlying mathematical principles. So, let's plug in our solution and give ourselves a pat on the back for a job well done!

Let's substitute $x = 7$ into the original equation:

$$3(7+9)^{\frac{3}{4}} = 24$$

$$3(16)^{\frac{3}{4}} = 24$$

Now, we simplify $(16)^{\frac{3}{4}}$. The fourth root of 16 is 2, since $2^4 = 16$. So,

$$(16)^{\frac{3}{4}} = (16^{\frac{1}{4}})^3 = 2^3$$

$$2^3 = 2 \cdot 2 \cdot 2 = 8$$

Substitute this back into the equation:

$$3(8) = 24$$

$$24 = 24$$

Our solution checks out!

Conclusion

So, the solution to the equation $3(x+9)^{\frac{3}{4}}=24$ is $x = 7$. Wasn't that fun? We tackled a tricky-looking equation by breaking it down into manageable steps. Remember, the key is to isolate the exponential term, eliminate the fractional exponent, simplify, and then isolate x. And always, always check your solution! Understanding these steps will empower you to solve a wide range of similar mathematical problems. You've now equipped yourself with a valuable tool in your mathematical arsenal. Keep practicing, and you'll become even more confident in your problem-solving abilities. Keep up the great work, and happy solving! This methodical approach, combined with consistent practice, will help you conquer any mathematical challenge that comes your way. Remember, every problem solved is a step forward in your mathematical journey. Keep exploring, keep learning, and keep having fun with math!