Solving (3/(3x-2))² * ³√(1/9) = 1 Find X + 1/3 Math Solution
Hey everyone! Let's dive into solving this interesting equation together. It looks a bit intimidating at first, but don't worry, we'll break it down step by step and make it super clear. Our mission is to find the value of 'x' that satisfies the equation (3/(3x-2))² * ³√(1/9) = 1. Not just that, we're also going to calculate x + 1/3 once we've nailed down the value of x. So, buckle up, and let's get started!
Understanding the Equation
First things first, let's really understand what this equation is telling us. We've got a fraction being squared, a cube root hanging out, and all of it equals one. The key here is to remember our order of operations (PEMDAS/BODMAS) and the properties of exponents and radicals.
Let's rewrite the equation a bit to make things clearer:
(3/(3x-2))² * ³√(1/9) = 1
This can also be seen as:
(3² / (3x-2)²) * (1/9)^(1/3) = 1
Breaking it down, we have:
- A fraction 3 divided by (3x-2), and the whole thing is squared. This means both the numerator (3) and the denominator (3x-2) are squared.
- Then we're multiplying by the cube root of 1/9. Remember, a cube root is the same as raising something to the power of 1/3.
- And all of this should equal 1. This tells us that whatever we have on the left side, after all the operations, it simplifies to 1. That's a crucial piece of information!
When we see equations like this, it’s important to not panic, guys! Just take it one piece at a time. Think about the properties and rules you know, and how they can help simplify the equation. Now, let’s move on to the next step: simplifying things!
Simplifying the Equation
Now comes the fun part – simplifying! Our goal here is to make the equation look less scary and more manageable. We're going to use some exponent rules and properties of radicals to clean things up. This is where we separate the pros from the Joes, so pay close attention!
Remember how we rewrote the equation earlier? Let's start there:
(3² / (3x-2)²) * (1/9)^(1/3) = 1
The first thing we can do is simplify 3². That's just 3 times 3, which equals 9. So, we have:
(9 / (3x-2)²) * (1/9)^(1/3) = 1
Next up is dealing with that cube root. We have (1/9)^(1/3). Now, 9 is the same as 3², so 1/9 can be written as 1/3². This is where knowing your powers and roots really helps!
So, (1/9)^(1/3) is the same as (1/3²)^(1/3). Using the rule of exponents that says (am)n = a^(m*n), we can simplify this further:
(1/3²)^(1/3) = 1/3^(2*(1/3)) = 1/3^(2/3)
Now our equation looks like this:
(9 / (3x-2)²) * (1/3^(2/3)) = 1
We're getting there, guys! See how breaking things down makes them less intimidating? We've simplified the square and the cube root. Now, let's move on to isolating the term with 'x' in it. This is where we use some algebraic manipulation to get the 'x' term by itself.
Isolating the Term with 'x'
Alright, let's get that 'x' term all by its lonesome on one side of the equation! To do this, we're going to use some algebraic magic – specifically, multiplying and dividing to get rid of the stuff we don't want. This is where a solid understanding of algebra really shines, so make sure you've got your fundamentals down!
We're starting with:
(9 / (3x-2)²) * (1/3^(2/3)) = 1
The first thing we can do is get rid of that (1/3^(2/3)) term. Since it's being multiplied, we can multiply both sides of the equation by its reciprocal, which is 3^(2/3). Remember, whatever you do to one side, you gotta do to the other!
Multiplying both sides by 3^(2/3) gives us:
(9 / (3x-2)²) = 3^(2/3)
Now, we want to get the (3x-2)² term out of the denominator. We can do this by multiplying both sides by (3x-2)²:
9 = 3^(2/3) * (3x-2)²
Looking good! Now, let's isolate (3x-2)² by dividing both sides by 3^(2/3):
9 / 3^(2/3) = (3x-2)²
We can simplify the left side a bit. Remember that 9 is 3², so we have:
3² / 3^(2/3) = (3x-2)²
Using the rule for dividing exponents with the same base (a^m / a^n = a^(m-n)), we get:
3^(2 - 2/3) = (3x-2)²
2 - 2/3 is 4/3, so we have:
3^(4/3) = (3x-2)²
Now, we've got the term with 'x' squared. The next step is to get rid of that square. We can do this by taking the square root of both sides. But hold on a sec... this is a crucial point! When we take the square root, we need to remember that there are potentially two solutions: a positive one and a negative one. This is because both the positive and negative square root, when squared, will give you the same result.
Solving for 'x'
Okay, we're in the home stretch now! We've simplified the equation, isolated the (3x-2)² term, and now it's time to actually solve for 'x'. Remember that crucial point we talked about – the one about the plus and minus when taking the square root? That's going to be super important here.
We're at this point in the equation:
3^(4/3) = (3x-2)²
Let's take the square root of both sides, remembering our plus and minus:
±√(3^(4/3)) = 3x - 2
We can simplify √(3^(4/3)). Remember that the square root is the same as raising something to the power of 1/2. So, we have:
√(3^(4/3)) = (3(4/3))(1/2)
Using our exponent rules again, (am)n = a^(m*n), we get:
(3(4/3))(1/2) = 3^((4/3)*(1/2)) = 3^(2/3)
So now we have:
±3^(2/3) = 3x - 2
This gives us two separate equations to solve:
- 3^(2/3) = 3x - 2
- -3^(2/3) = 3x - 2
Let's solve each one individually. For the first equation, 3^(2/3) = 3x - 2, we add 2 to both sides:
3^(2/3) + 2 = 3x
Then, we divide both sides by 3:
x = (3^(2/3) + 2) / 3
That's one solution for 'x'! Now, let's tackle the second equation, -3^(2/3) = 3x - 2. Again, we add 2 to both sides:
-3^(2/3) + 2 = 3x
And divide by 3:
x = (-3^(2/3) + 2) / 3
And there's our second solution! We have two possible values for 'x'. Give yourselves a pat on the back, guys – this was a big step!
Calculating x + 1/3
We've conquered the tricky part – finding the values of 'x'! Now, it's time for the cherry on top: calculating x + 1/3 for each of our solutions. This is a pretty straightforward calculation once we have the values of 'x', but let’s make sure we get it right. Accuracy is key in mathematics, just like hitting the right note in music, you know?
Remember, we found two solutions for 'x':
- x = (3^(2/3) + 2) / 3
- x = (-3^(2/3) + 2) / 3
Let's calculate x + 1/3 for each of these. For the first solution:
x + 1/3 = ((3^(2/3) + 2) / 3) + 1/3
To add these fractions, we need a common denominator, which we already have – it's 3! So, we can just add the numerators:
x + 1/3 = (3^(2/3) + 2 + 1) / 3
Simplify the numerator:
x + 1/3 = (3^(2/3) + 3) / 3
And that's our first value for x + 1/3!
Now, let's do the same for the second solution:
x + 1/3 = ((-3^(2/3) + 2) / 3) + 1/3
Again, we have a common denominator, so we add the numerators:
x + 1/3 = (-3^(2/3) + 2 + 1) / 3
Simplify:
x + 1/3 = (-3^(2/3) + 3) / 3
And that's our second value for x + 1/3! We've done it! We found the values of 'x', and we calculated x + 1/3 for each of them. Give yourselves a round of applause, folks!
Final Answer
Alright, guys, we've reached the end of our mathematical journey! Let's recap what we've accomplished and present our final answer in a clear and concise way. We started with a seemingly complex equation: (3/(3x-2))² * ³√(1/9) = 1, and we systematically broke it down, simplified it, and solved it like pros. Remember, the key is to take things one step at a time, use the properties and rules you know, and don't be afraid to get your hands dirty with some algebra!
We found two solutions for 'x':
- x = (3^(2/3) + 2) / 3
- x = (-3^(2/3) + 2) / 3
And then, we calculated x + 1/3 for each solution:
- x + 1/3 = (3^(2/3) + 3) / 3
- x + 1/3 = (-3^(2/3) + 3) / 3
So, these are our final answers! We've successfully solved the equation and calculated the desired expression. I hope you found this breakdown helpful and that it made the process a bit clearer. Remember, math is like a puzzle – it might seem tough at first, but with the right approach and a little bit of practice, you can solve anything!
If you have any questions or want to tackle more problems like this, feel free to ask! Keep practicing, keep exploring, and most importantly, keep having fun with math!
