Solving The Exponential Equation: (9/√3)^(2x) = 1/81

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Hey guys! Today, we're diving into a fun little math problem that involves solving an exponential equation. Specifically, we're going to tackle the equation (9/√3)^(2x) = 1/81. Don't worry if it looks intimidating at first; we'll break it down step by step so it's super easy to understand. Let's get started!

Understanding Exponential Equations

Before we jump into solving, let's quickly recap what exponential equations are all about. An exponential equation is basically an equation where the variable appears in the exponent. Think of it like this: instead of solving for something like x in 2x = 4, we're solving for x in something like 2^x = 4. The key to solving these equations is to manipulate them so that we can compare exponents directly. This often involves expressing both sides of the equation with the same base. This allows us to equate the exponents and solve for our variable.

Why is this important? Well, exponential equations pop up all over the place in real life! They're used to model things like population growth, radioactive decay, compound interest in finance, and even the spread of diseases. So, mastering the art of solving them is a pretty valuable skill to have in your mathematical toolkit. In this particular problem, we will need to simplify expressions, use exponent rules, and solve a simple linear equation. So, let’s keep these concepts in mind as we proceed!

Step-by-Step Solution

Now, let's get our hands dirty with the problem at hand: (9/√3)^(2x) = 1/81. We'll go through each step methodically to make sure we don't miss anything.

1. Simplify the Base on the Left Side

The first thing we want to do is simplify the base on the left side of the equation, which is (9/√3). This looks a bit messy, so let's clean it up. We can rewrite 9 as 3^2, so we have (3^2/√3). Now, remember that √3 is the same as 3^(1/2). So, we can rewrite our expression as (32)/(3(1/2)).

To simplify this further, we use the rule of exponents that says when you divide powers with the same base, you subtract the exponents. In other words, a^m / a^n = a^(m-n). Applying this rule, we get 3^(2 - 1/2) = 3^(3/2). So, we've successfully simplified (9/√3) to 3^(3/2). Doesn't that look much nicer?

2. Rewrite the Right Side with the Same Base

Next up, we need to tackle the right side of the equation, which is 1/81. We want to express this as a power of 3, just like we did with the left side. We know that 81 is 3^4 (3 * 3 * 3 * 3 = 81). So, 1/81 can be written as 1/(3^4). Now, remember another handy rule of exponents: 1/a^n = a^(-n). Using this rule, we can rewrite 1/(3^4) as 3^(-4). Awesome!

3. Substitute the Simplified Expressions Back into the Equation

Now that we've simplified both sides, let's plug them back into our original equation. We had (9/√3)^(2x) = 1/81. We simplified (9/√3) to 3^(3/2) and 1/81 to 3^(-4). So, our equation now looks like this: (3(3/2))(2x) = 3^(-4). See how much cleaner it is already?

4. Simplify the Left Side Further

We're not quite done simplifying the left side. We have (3(3/2))(2x). Remember the rule of exponents that says when you raise a power to another power, you multiply the exponents? That is, (am)n = a^(m*n). Applying this rule, we get 3^((3/2) * 2x) = 3^(3x). So, our equation now looks like this: 3^(3x) = 3^(-4).

5. Equate the Exponents

This is the crucial step! Now that we have the same base (3) on both sides of the equation, we can simply equate the exponents. If a^m = a^n, then m = n. So, in our case, we have 3^(3x) = 3^(-4), which means 3x = -4. We're almost there!

6. Solve for x

Finally, we have a simple linear equation to solve for x: 3x = -4. To isolate x, we just divide both sides by 3. This gives us x = -4/3. And that's our answer! We've successfully solved the exponential equation.

Solution Breakdown

Let's quickly recap what we did:

  1. Simplified the base on the left side: (9/√3) became 3^(3/2).
  2. Rewrote the right side with the same base: 1/81 became 3^(-4).
  3. Substituted the simplified expressions back into the equation: (3(3/2))(2x) = 3^(-4).
  4. Simplified the left side further: 3^((3/2) * 2x) = 3^(3x).
  5. Equated the exponents: 3x = -4.
  6. Solved for x: x = -4/3.

So, the solution to the equation (9/√3)^(2x) = 1/81 is x = -4/3. Great job!

Common Mistakes to Avoid

When solving exponential equations, there are a few common pitfalls to watch out for. Let's go over them so you can steer clear.

Forgetting the Exponent Rules

Exponent rules are the bread and butter of solving these types of equations. Forgetting or misapplying them can lead to major headaches. Make sure you have a solid grasp of rules like a^m / a^n = a^(m-n), 1/a^n = a^(-n), and (am)n = a^(m*n). Keep these in your back pocket, and you'll be well-equipped to tackle any exponential equation that comes your way.

Not Simplifying Correctly

Simplifying expressions is a crucial step. If you don't simplify the bases and exponents properly, you'll end up with a much more complicated equation to solve. Always try to express both sides of the equation with the same base before equating the exponents. It makes life so much easier!

Making Arithmetic Errors

Simple arithmetic mistakes can throw off your entire solution. Double-check your calculations, especially when dealing with fractions and negative signs. It’s easy to make a small slip-up, but catching it early can save you a lot of frustration.

Not Checking the Solution

It's always a good idea to plug your solution back into the original equation to make sure it works. This helps you catch any errors you might have made along the way. Just substitute your value of x back into the original equation and see if both sides are equal. If they are, you're golden!

Practice Problems

To really nail down your skills, let's try a few practice problems. Remember, practice makes perfect! Work through these on your own, and then check your answers. You'll be an exponential equation whiz in no time.

  1. Solve for x: 2^(3x) = 16
  2. Solve for x: (1/5)^(2x) = 25
  3. Solve for x: 4^(x+1) = 8

Conclusion

So, guys, we've walked through how to solve the exponential equation (9/√3)^(2x) = 1/81 step by step. We covered simplifying bases, using exponent rules, and equating exponents. Remember, the key is to break the problem down into manageable chunks and apply the rules of exponents systematically. With a bit of practice, you'll be solving these equations like a pro! Keep practicing, and don't be afraid to tackle more challenging problems. You've got this! Remember to watch out for those common mistakes and always double-check your work. Happy solving!