Hey guys! Today, we're diving into a super common type of math problem: solving equations with cube roots. Don't worry, it's not as scary as it sounds! We'll break it down step-by-step, so you'll be a pro in no time. Our specific problem is: $\sqrt[3]{4 x-3}+4=7$, and we're going to solve for x. Let's get started!
Understanding Cube Roots
Before we jump into the equation, let's quickly refresh our understanding of cube roots. Think of a cube root as the opposite of cubing a number. For example, the cube root of 8 is 2 because 2 cubed (2 * 2 * 2) equals 8. Similarly, the cube root of 27 is 3 because 3 cubed is 27. This understanding is crucial for solving equations involving cube roots. We're essentially trying to 'undo' the cubing operation to isolate our variable, x. Keep this in mind as we move forward, because this core concept will guide our steps in solving the equation. We need to isolate the cube root term, then eliminate it by cubing both sides. This might sound a bit abstract now, but it will become clear as we work through the example. Remember, the key is to think about inverse operations – just like subtraction undoes addition, and division undoes multiplication, taking the cube root undoes cubing. This process of 'undoing' operations is a fundamental principle in algebra, so mastering it here will definitely help you in tackling other types of equations as well. And remember, practice makes perfect! The more you work with cube roots, the more comfortable you'll become with them. Don't be afraid to try different examples and play around with numbers to solidify your understanding. Trust me, once you get the hang of it, you'll feel a real sense of accomplishment!
Step 1: Isolate the Cube Root
Our first goal is to get the cube root term, $\sqrt[3]{4 x-3}$, all by itself on one side of the equation. Currently, we have a '+ 4' hanging out with it. To get rid of this '+ 4', we need to perform the inverse operation, which is subtraction. We'll subtract 4 from both sides of the equation to maintain balance. Remember, what we do to one side, we must do to the other! This is a golden rule in algebra, and it's super important for ensuring that our equation remains equal. If we only subtracted 4 from one side, the equation would become unbalanced, and our solution would be incorrect. So, let's go ahead and subtract 4 from both sides. On the left side, the '+ 4' and '- 4' will cancel each other out, leaving us with just the cube root term. On the right side, we'll have 7 - 4, which is a simple subtraction. This step is a perfect example of how we use inverse operations to isolate the variable we're trying to solve for. It's like peeling back layers to reveal the hidden value of x. And by isolating the cube root, we're setting ourselves up for the next crucial step: eliminating the cube root altogether. So, keep that balance in mind, and remember that each step we take is a deliberate move towards our final solution.
So, we have:
Subtracting 4 from both sides:
This simplifies to:
Step 2: Eliminate the Cube Root
Now that we have the cube root isolated, it's time to get rid of it! How do we do that? Well, remember what we said about cube roots and cubing being inverse operations? That's exactly what we'll use here. To eliminate the cube root, we need to cube both sides of the equation. Cubing a cube root cancels it out, leaving us with just the expression inside the cube root. This is where the power of inverse operations really shines. By cubing both sides, we're effectively undoing the cube root operation, which allows us to access the expression inside – in this case, 4x - 3. But remember, just like in the previous step, we must cube both sides to maintain the equality. If we only cubed one side, the equation would become unbalanced, and we'd be back to square one. So, let's cube both sides with confidence, knowing that this is the key to unlocking the value of x. Cubing both sides might seem like a big step, but it's a logical progression in our solution. We've isolated the cube root, and now we're using the inverse operation to eliminate it. This is a common strategy in algebra, and it's one you'll use again and again. So, let's cube away and see what we get!
So, cubing both sides of $\\sqrt[3]{4 x-3}=3$ gives us:
Which simplifies to:
Step 3: Isolate the Variable Term
We're making great progress! We've eliminated the cube root, and now we have a much simpler equation: 4x - 3 = 27. Our next goal is to isolate the term with x in it, which is 4x. To do this, we need to get rid of the '- 3' that's hanging out with it. Just like before, we'll use the inverse operation. The opposite of subtracting 3 is adding 3, so we'll add 3 to both sides of the equation. Remember the golden rule: what you do to one side, you must do to the other! This is crucial for keeping the equation balanced and ensuring that our solution is correct. Adding 3 to both sides will effectively cancel out the '- 3' on the left side, leaving us with just 4x. On the right side, we'll simply add 3 to 27. This step is another example of how we strategically use inverse operations to isolate the variable. We're systematically peeling away the layers surrounding x until we can finally reveal its value. Each step brings us closer to the solution, and by following these logical steps, we're building a solid foundation for solving more complex equations in the future. So, let's add 3 to both sides and see what our equation looks like now!
Adding 3 to both sides of 4x - 3 = 27 gives us:
Which simplifies to:
Step 4: Solve for x
We're in the home stretch! We've got 4x = 30. Now, we just need to get x all by itself. The '4' is currently multiplying x. To undo multiplication, we use division. So, we'll divide both sides of the equation by 4. You guessed it – remember the golden rule! Dividing both sides by 4 will isolate x on the left side. On the right side, we'll have 30 divided by 4, which we can simplify. This final step is the culmination of all our hard work. We've strategically used inverse operations to peel away all the layers surrounding x, and now we're ready to reveal its value. Dividing both sides by 4 is the final piece of the puzzle, and it will give us the solution we've been searching for. So, let's divide and conquer, and find the value of x!
Dividing both sides of 4x = 30 by 4 gives us:
This simplifies to:
Or, as a decimal:
Answer
Therefore, the solution to the equation $\sqrt[3]{4 x-3}+4=7$ is $x = \frac{15}{2}$ or 7.5.
Conclusion
And there you have it! We've successfully solved the equation $\sqrt[3]{4 x-3}+4=7$ by following a step-by-step approach. We isolated the cube root, eliminated it by cubing both sides, isolated the variable term, and finally solved for x. Remember the key concepts: inverse operations and maintaining balance in the equation. By mastering these principles, you'll be able to tackle a wide range of algebraic equations with confidence. So, keep practicing, and don't be afraid to challenge yourself! You've got this!