Solving For M: (1/27)^m * (81)^(-1) = 243 - Step-by-Step Guide

Hey guys! Let's dive into this math problem together. We're going to figure out how to find the value of 'm' in the equation (1/27)^m * (81)^(-1) = 243. Don't worry, we'll break it down step by step so it's super easy to follow. Let’s get started!

Understanding the Basics

Before we jump into solving for m, it’s important to understand the basic concepts we’ll be using. This problem mainly involves exponents and powers. Let's make sure we're all on the same page.

• Exponents: An exponent tells you how many times a number (the base) is multiplied by itself. For example, in 2^3, 2 is the base and 3 is the exponent. This means 2 * 2 * 2 = 8.
• Negative Exponents: A negative exponent means you take the reciprocal of the base raised to the positive exponent. So, x^(-n) is the same as 1 / (x^n).
• Fractional Bases: When you have a fraction as a base, like (1/2)^2, you raise both the numerator and the denominator to the exponent. So, (1/2)^2 = 1^2 / 2^2 = 1/4.
• Expressing Numbers as Powers of a Common Base: This is a crucial technique for solving exponential equations. If you can express all numbers in the equation as powers of the same base, you can then equate the exponents. For example, both 4 and 8 can be expressed as powers of 2 (4 = 2^2 and 8 = 2^3).

Knowing these basics will help us tackle the problem with confidence. Remember, math is like building blocks – each concept builds on the previous one. So, understanding these fundamentals is key to mastering more complex problems. Now, let's see how we can apply these ideas to solve our equation!

Breaking Down the Equation

Okay, so our main goal here is to find the value of m in the equation:

(1/27)^m * (81)^(-1) = 243

The secret to cracking this is to express all these numbers as powers of a common base. In this case, the magic number is 3. Why? Because 27, 81, and 243 are all powers of 3. Let's break it down:

• 27 = 3 * 3 * 3 = 3^3
• 81 = 3 * 3 * 3 * 3 = 3^4
• 243 = 3 * 3 * 3 * 3 * 3 = 3^5

Now, let's rewrite our equation using these powers of 3. This is where things start to get interesting!

(1/27)^m becomes (1/3³⁾m, which is the same as (3^(-3))m. Remember, 1/x^n = x^(-n)? We're using that trick here.

(81)^(-1) becomes (3⁴⁾(-1), which is simply 3^(-4). We're just substituting 81 with its equivalent, 3^4, and keeping the exponent of -1.

So, 243 is just 3^5, as we figured out earlier.

Now, let’s put it all together. Our original equation (1/27)^m * (81)^(-1) = 243 can now be written as:

(3^(-3))m * 3^(-4) = 3^5

See how we've transformed the equation into something much simpler? By expressing everything as powers of 3, we've set ourselves up for the next step: simplifying those exponents. Stick with me, guys – we're getting closer to the solution!

Simplifying the Exponents

Alright, now that we've got our equation in terms of powers of 3, it's time to simplify those exponents. Remember the rule that says (x^a)b = x^(a*b)? This is going to be our best friend here. Let's apply it to our equation:

(3^(-3))m * 3^(-4) = 3^5

First, let's tackle (3^(-3))m. Using our rule, this simplifies to 3^(-3m). We're just multiplying the exponents -3 and m.

Now our equation looks like this:

3^(-3m) * 3^(-4) = 3^5

Next up, we've got 3^(-3m) multiplied by 3^(-4). Here’s another handy exponent rule: x^a * x^b = x^(a+b). When you're multiplying powers with the same base, you add the exponents. So, we add -3m and -4.

This gives us:

3^(-3m - 4) = 3^5

Look at that! We've managed to condense the left side of the equation into a single power of 3. Now we've got the equation in a super manageable form. What’s the big deal? Well, now we can use a really cool trick to solve for m. Stay tuned!

Equating the Exponents

Okay, guys, this is where the magic really happens! We’ve simplified our equation to this:

3^(-3m - 4) = 3^5

Notice something cool? Both sides of the equation are now expressed as powers of the same base, which is 3. This is huge because it means we can equate the exponents. If 3 raised to one power equals 3 raised to another power, then those powers must be equal. Make sense?

So, we can take the exponents from both sides and set them equal to each other. This gives us a simple equation:

-3m - 4 = 5

See how we've transformed a tricky exponential equation into a basic linear equation? This is the power of simplifying and using those exponent rules we talked about earlier. Now, all that's left is to solve this equation for m. We're in the home stretch now!

Solving for 'm'

Alright, let's finish this! We've got our simplified equation:

-3m - 4 = 5

Now it’s just a matter of using some basic algebra to isolate m. First, we want to get the term with m by itself on one side of the equation. To do this, we add 4 to both sides. This cancels out the -4 on the left side:

-3m - 4 + 4 = 5 + 4

Which simplifies to:

-3m = 9

Looking good! Now, we need to get m all by itself. It's currently being multiplied by -3, so we need to do the opposite operation: divide both sides by -3:

-3m / -3 = 9 / -3

This gives us:

m = -3

Boom! We did it! We found the value of m. It’s -3. Awesome work, guys! See, tackling these problems step by step makes them way less intimidating.

Final Answer

So, after breaking down the equation (1/27)^m * (81)^(-1) = 243 and going through all the steps, we've arrived at our final answer:

m = -3

To recap, here’s what we did:

1. Expressed all numbers as powers of 3.
2. Simplified the exponents using exponent rules.
3. Equated the exponents.
4. Solved the resulting linear equation for m.

By following these steps, we turned a seemingly complex problem into a manageable one. Remember, the key to mastering math is understanding the basic concepts and applying them systematically. Keep practicing, and you'll become a pro at solving these types of equations. Great job, everyone!