Solve For N In The Exponential Equation (216^(n-2))/((1/36)^(3n)) = 216

Hey there, math enthusiasts! Let's dive into an exciting exponential equation problem today. We've got a fascinating challenge on our hands: finding the value of 'n' that satisfies the equation $\frac{216^{{n-2}}{\left(\frac{1}{36}\right)}{3 n}}=216 $. Don't worry if it looks a bit intimidating at first. We're going to break it down step by step, making sure everyone can follow along. So, grab your pencils and let's get started!

Let's break down the problem step by step

Understanding the Basics of Exponential Equations

Before we jump into solving this specific equation, let's quickly recap what exponential equations are all about. In essence, exponential equations involve variables in the exponents. Our primary goal is to isolate the variable, but with exponents, we often need to manipulate the equation using exponent rules and properties. The core idea is to express both sides of the equation with the same base. Once we achieve that, we can equate the exponents and solve for our variable. Remember those exponent rules from algebra? They're our best friends here! For example, recall the rule $(a^m)^n = a^{mn}$ and $\frac{a^m}{a^n} = a^{m-n}$. These will be instrumental in simplifying our equation. Another crucial concept is expressing numbers as powers of a common base. Notice that 216 and 36 are both related to 6 (since $6^3 = 216$ and $6^2 = 36$). This observation will be key to solving the problem efficiently. So, with our basic toolkit ready, we are now fully equipped to tackle the problem at hand and demystify this exponential puzzle. We'll start by rewriting the given equation in terms of a common base, making sure each step is crystal clear.

Rewriting the Equation with a Common Base

Okay, so let's tackle our equation: $\frac216^{{n-2}}{\left(\frac{1}{36}\right)}{3 n}}=216 $. The first thing we need to do is express all the terms with a common base. As we noticed earlier, both 216 and 36 are powers of 6. We know that $216 = 6^3$ and $36 = 6^2$. Also, remember that $\frac{1}{36}$ can be written as $6^{-2}$. Now, let’s substitute these into our equation. We get $\frac{(6³⁾{n-2}(6^{-2}){3n}} = 6^3$. See how we've replaced 216 with $6^3$ and $\frac{1}{36}$ with $6^{-2}$? This is a crucial step because it allows us to use exponent rules to simplify the equation further. Next, we'll apply the power of a power rule, which states that $(a^m)^n = a^{mn}$. Applying this rule, the equation becomes $\frac{6^{3(n-2)}{6^{-6n}} = 6^3$. We've simplified the exponents, and the equation is starting to look much cleaner. This transformation is key because it allows us to directly compare the exponents once we simplify the left side of the equation further. By using the common base of 6, we've laid the groundwork for solving the equation by focusing on the exponents.

Simplifying the Exponents

Now that we've rewritten the equation with a common base, let's simplify those exponents. Our equation currently looks like this: $\frac{6^{3(n-2)}}{6^{-6n}} = 6^3$. The next step involves using another exponent rule: $\frac{a^m}{a^n} = a^{m-n}$. Applying this rule to the left side of the equation, we subtract the exponents: $6^{3(n-2) - (-6n)} = 6^3$. This simplifies to $6^{3n - 6 + 6n} = 6^3$. Now, combine like terms in the exponent: $6^{9n - 6} = 6^3$. We're getting closer! Notice how the equation is becoming more manageable. By using exponent rules to simplify, we've reduced the complexity and set ourselves up for the final step: equating the exponents. The exponent on the left side, $9n - 6$, now needs to be equal to the exponent on the right side, which is 3. This sets up a simple linear equation that we can solve for 'n'. Keep pushing forward, guys; we're almost there!

Solving for n

Alright, guys, we're in the home stretch! We've simplified our equation to: $6^{9n - 6} = 6^3$. Now comes the most exciting part – solving for 'n'. Since the bases are the same (both are 6), we can equate the exponents. This means we set the exponent on the left side equal to the exponent on the right side: $9n - 6 = 3$. See how the exponential equation has transformed into a simple linear equation? Now, let's solve this linear equation. First, add 6 to both sides: $9n = 3 + 6$, which simplifies to $9n = 9$. Then, divide both sides by 9: $n = \frac{9}{9}$, which gives us $n = 1$. And there we have it! We've found the value of 'n' that satisfies the original equation. It's a fantastic feeling when everything comes together like this, isn't it? We started with a complex-looking exponential equation, used exponent rules to simplify, and ended up with a straightforward solution. So, the value of 'n' that makes the equation true is 1.

Checking Our Solution

Verifying the Solution

Before we celebrate our victory, it’s always a good idea to double-check our answer. Remember, we found that $n = 1$. Let's plug this value back into the original equation to make sure it holds true. The original equation was: $\frac216^{{n-2}}{\left(\frac{1}{36}\right)}{3 n}}=216 $. Now, substitute $n = 1$ into the equation $\frac{216^{1-2}\left(\frac{1}{36}\right)^{3 (1)}}=216 $. This simplifies to $\frac{216^{-1}\left(\frac{1}{36}\right)^{3}}=216 $. Let’s break this down further. We know that $216^{-1} = \frac{1}{216}$ and $\left(\frac{1}{36}\right)^{3} = \frac{1}{36^3}$. So, our equation becomes $\frac{\frac{1216}}{\frac{1}{36^3}}=216 $. To divide by a fraction, we multiply by its reciprocal $\frac{1216} \times 36^3 = 216 $. Now, let's rewrite $36^3$ as $(6^2)^3 = 6^6$. Also, recall that $216 = 6^3$. So, we have $\frac{16^3} \times 6^6 = 6^3 $. Simplify the left side $6^{6-3 = 6^3$, which gives us $6^3 = 6^3$. This is true! Our solution checks out. Plugging $n = 1$ into the original equation makes it a true statement, which means we've correctly solved for 'n'.

Conclusion

Wrapping Up Our Exponential Adventure

Great job, guys! We've successfully navigated through this exponential equation and found the value of 'n'. To recap, we started with the equation $\frac{216^{{n-2}}{\left(\frac{1}{36}\right)}{3 n}}=216 $ and were tasked with finding the value of 'n' that makes it true. We began by recognizing that 216 and 36 could both be expressed as powers of 6, which allowed us to rewrite the equation with a common base. This is a crucial step in solving exponential equations because it enables us to use exponent rules effectively. We then simplified the equation using these rules, such as $(a^m)^n = a^{mn}$ and $\frac{a^m}{a^n} = a^{m-n}$, ultimately transforming the equation into $6^{9n - 6} = 6^3$. Once we had the same base on both sides, we equated the exponents, leading us to the linear equation $9n - 6 = 3$. Solving for 'n' gave us $n = 1$. Finally, we verified our solution by plugging $n = 1$ back into the original equation, confirming that it holds true. This entire process highlights the power of breaking down complex problems into smaller, manageable steps. By applying the rules of exponents and simplifying strategically, we were able to demystify this equation and find the solution. Keep practicing these techniques, and you'll become a master at solving exponential equations! Remember, math is an adventure, and every problem solved is a step forward.

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Solving Exponential Equations Find n Value for (216^{(n-2))/((1/36)}(3n))=216