Solving For A In The Equation 9 = (1/27)^(a+3)
Hey guys! Let's dive into this interesting exponential equation: 9 = (1/27)^(a+3). Our mission, should we choose to accept it (and we totally do!), is to find the value of 'a' that makes this equation true. Don't worry, it's not as intimidating as it looks. We'll break it down step-by-step, making sure everyone understands the logic and math involved. So, buckle up, and let's get started!
Expressing Both Sides with the Same Base
The golden rule when tackling exponential equations is to try and express both sides with the same base. This simplifies the equation significantly, allowing us to equate the exponents. Looking at our equation, 9 = (1/27)^(a+3), we can see that both 9 and 1/27 can be expressed as powers of 3. This is a crucial observation! 9 is simply 3 squared (3^2), and 1/27 is 3 to the power of -3 (3^-3) because 27 is 3 cubed (3^3), and the reciprocal (1/27) makes the exponent negative. Why is this important, you ask? Well, expressing both sides with the same base allows us to directly compare the exponents. Think of it like this: if we have 3 raised to some power on one side and 3 raised to another power on the other side, and the bases (which are both 3) are equal, then the powers themselves must also be equal. It's a fundamental principle of exponential functions. So, let's rewrite our equation using 3 as the base:
3^2 = (3-3)(a+3)
This transformation is a key step. By expressing both sides with the same base, we've set the stage for the next step: simplifying the equation and isolating 'a'. Remember, the goal here is to manipulate the equation in a way that 'a' eventually stands alone on one side, revealing its value. We're essentially peeling away the layers of the equation, one operation at a time, until we get to the core: the value of 'a'. Now that we have both sides expressed with the same base, the power rule of exponents comes into play. This rule is our next weapon in this mathematical quest.
Applying the Power of a Power Rule
Now that we've rewritten the equation as 3^2 = (3-3)(a+3), let's bring in the power of a power rule! This rule states that when you raise a power to another power, you multiply the exponents. In simpler terms, (xm)n is the same as x^(mn). This is exactly what we need to simplify the right side of our equation. We have 3 raised to the power of -3, and that entire term is raised to the power of (a+3). Applying the power rule, we multiply -3 by (a+3). This gives us -3(a+3), which simplifies to -3a - 9. So, the right side of the equation becomes 3 raised to the power of (-3a - 9). Our equation now looks like this:
3^2 = 3^(-3a - 9)
See how things are getting clearer? We've successfully simplified the right side by applying the power rule. Both sides of the equation now have the same base (3), but they have different exponents. This is where the magic happens! Since the bases are the same, the exponents must be equal for the equation to hold true. It's like saying if 2^x = 2^y, then x must equal y. This is the fundamental principle that allows us to move from an exponential equation to a simple algebraic equation. The exponential part is essentially 'gone' now, and we're left with a linear equation that we can easily solve. So, let's equate the exponents and set up our next step towards finding 'a'. This step is where the equation transforms from something a bit intimidating into a straightforward algebraic problem. It's like we've cracked the code, and now we just need to follow the instructions to open the treasure chest (which, in this case, contains the value of 'a').
Equating the Exponents
Alright, we've arrived at a crucial point! We have the equation 3^2 = 3^(-3a - 9). As we discussed, because the bases are the same (both are 3), we can now equate the exponents. This means the exponent on the left side (which is 2) must be equal to the exponent on the right side (which is -3a - 9). This gives us a simple linear equation:
2 = -3a - 9
Isn't that neat? We've transformed a potentially scary exponential equation into a friendly, solvable algebraic equation. Now, it's just a matter of isolating 'a'. This is like unwrapping a present – each step gets us closer to the surprise inside. Solving for 'a' in this linear equation involves a few basic algebraic manipulations. The goal is to get 'a' all by itself on one side of the equation. We can do this by adding 9 to both sides and then dividing by -3. These are standard algebraic techniques that we use all the time to solve for unknowns. It's like following a recipe – we know the steps, and we just need to execute them in the correct order to get the desired result. This step is a testament to the power of simplification in mathematics. By using the properties of exponents, we've made the problem much easier to handle. So, let's get those algebraic gears turning and isolate 'a'.
Isolating 'a' and Solving the Equation
Let's continue our quest to find the value of 'a'! We have the equation 2 = -3a - 9. To isolate 'a', our first step is to get rid of the -9 on the right side. We can do this by adding 9 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. This is a fundamental principle of algebra – maintaining equality. Adding 9 to both sides gives us:
2 + 9 = -3a - 9 + 9
This simplifies to:
11 = -3a
Now, we're one step closer! We have -3 multiplied by 'a' on the right side. To isolate 'a', we need to get rid of the -3. The opposite of multiplication is division, so we'll divide both sides of the equation by -3. Again, maintaining balance is key. Dividing both sides by -3 gives us:
11 / -3 = -3a / -3
This simplifies to:
-11/3 = a
Or, we can write it as:
a = -11/3
And there we have it! We've successfully isolated 'a' and found its value. It's like reaching the summit of a mountain after a challenging climb – the view is definitely worth the effort. The value of 'a' that satisfies the equation 9 = (1/27)^(a+3) is -11/3. We've gone from an exponential equation to a single numerical value, thanks to our understanding of exponents and algebraic manipulation. Now, to be absolutely sure, let's verify our solution. It's always a good idea to double-check our work, especially in mathematics, to ensure we haven't made any mistakes along the way. So, let's plug our value of 'a' back into the original equation and see if it holds true.
Verifying the Solution
Time to put our solution to the test! We found that a = -11/3, and we want to make sure this value satisfies the original equation: 9 = (1/27)^(a+3). To verify, we'll substitute -11/3 for 'a' in the equation and see if both sides are equal. This is like a final exam for our solution – if it passes, we know we've done everything correctly. Substituting a = -11/3 into the equation gives us:
9 = (1/27)^((-11/3)+3)
Now, let's simplify the exponent. We need to add -11/3 and 3. To do this, we need a common denominator. We can rewrite 3 as 9/3. So, the exponent becomes:
(-11/3) + (9/3) = -2/3
Our equation now looks like this:
9 = (1/27)^(-2/3)
Let's rewrite 1/27 as 3^-3, as we did earlier. This gives us:
9 = (3-3)(-2/3)
Now, we apply the power of a power rule again: multiply the exponents -3 and -2/3. This gives us:
(-3) * (-2/3) = 2
So, the equation becomes:
9 = 3^2
And, of course, 3^2 is equal to 9! So, we have:
9 = 9
This is a true statement! Our solution, a = -11/3, satisfies the original equation. We've not only solved the equation, but we've also verified our solution. This is a great feeling – like completing a puzzle and seeing all the pieces fit perfectly. We can confidently say that the value of 'a' that makes the equation true is -11/3. Now, let's recap what we've done and solidify our understanding.
Conclusion
So, guys, we've successfully navigated the world of exponential equations and found the value of 'a' in the equation 9 = (1/27)^(a+3). We started by expressing both sides of the equation with the same base (3), which allowed us to equate the exponents. Then, we solved the resulting linear equation for 'a', finding that a = -11/3. Finally, we verified our solution by plugging it back into the original equation and confirming that it holds true. This journey involved several key steps and concepts, including understanding exponential functions, applying the power of a power rule, and using basic algebraic manipulations. It's a testament to how mathematics builds upon itself – each concept and technique we learn becomes a tool in our problem-solving arsenal. Remember, the key to mastering these types of problems is practice and a solid understanding of the underlying principles. Don't be afraid to break down complex problems into smaller, more manageable steps. And always, always verify your solutions! Keep practicing, keep exploring, and keep the mathematical fires burning! You've got this!