Find Integers M And N Satisfying (2xⁿy²)ᵐ = 4x⁶y⁴

Hey guys! Today, we're diving into an exciting algebra problem where we need to find two integers, m and n, that make the equation (2xⁿy²)ᵐ = 4x⁶y⁴ true. This kind of problem is super cool because it combines exponents and algebraic manipulation, giving us a real workout for our brains. Let’s break it down step by step!

Understanding the Problem

First off, let's make sure we really understand what the question is asking. We've got an equation: (2xⁿy²)ᵐ = 4x⁶y⁴. Our mission, should we choose to accept it, is to find the values of m and n that make this equation a perfect match. Think of it like fitting puzzle pieces together; we need to find the right m and n to make both sides of the equation identical. This means not just getting the numbers right, but also making sure the exponents of x and y are spot on. So, let's get our algebraic gears turning and figure out how to crack this!

Step 1: Simplify the Left Side of the Equation

Okay, so the first thing we're gonna do is simplify the left side of the equation. We've got (2xⁿy²)ᵐ, and we need to distribute that exponent m to everything inside the parentheses. Remember the power of a product rule? It says that (ab)ᶜ = aᶜbᶜ. This means we need to apply the exponent m to the 2, the xⁿ, and the y². Let's do it:

(2xⁿy²)ᵐ = 2ᵐ * (xⁿ)ᵐ * (y²)ᵐ

Now, we need to simplify further. We have (xⁿ)ᵐ and (y²)ᵐ. Another exponent rule comes to our rescue: the power of a power rule, which states that (aᵇ)ᶜ = aᵇᶜ. Applying this rule, we get:

2ᵐ * (xⁿ)ᵐ * (y²)ᵐ = 2ᵐ * xⁿᵐ * y²ᵐ

Awesome! We've now transformed the left side into something much more manageable: 2ᵐ * xⁿᵐ * y²ᵐ. This simplified form is crucial because it allows us to directly compare the exponents and coefficients with the right side of the equation. Remember, our goal is to make the left side look exactly like the right side, so this is a major step forward. Next, we'll tackle the right side and see how we can match things up.

Step 2: Analyze the Right Side of the Equation

Alright, let's turn our attention to the right side of the equation, which is 4x⁶y⁴. This side looks simpler than the left side did initially, but we still need to think about it carefully. Specifically, we need to recognize that the number 4 can be expressed as a power of 2. Why is this important? Well, if you remember from our simplification of the left side, we ended up with 2 raised to the power of m (2ᵐ). To make the two sides equal, we're likely going to need to match that 2ᵐ with the 4 on the right side.

So, let's rewrite 4 as 2². Now our right side looks like this: 2² * x⁶ * y⁴. See how things are starting to line up? We have a power of 2, an x term with an exponent, and a y term with an exponent. This is exactly what we have on the left side after simplification: 2ᵐ * xⁿᵐ * y²ᵐ. The key now is to equate the corresponding parts and solve for m and n. We're in the home stretch, guys!

Step 3: Equate Coefficients and Exponents

Okay, the most exciting part! Now we need to equate the coefficients and exponents from both sides of our equation. We’ve simplified both sides, so we have:

Left side: 2ᵐ * xⁿᵐ * y²ᵐ Right side: 2² * x⁶ * y⁴

To make these two sides equal, each corresponding part must be equal. This gives us a system of equations:

1. Matching the coefficients of 2: 2ᵐ = 2²
2. Matching the exponents of x: nm = 6
3. Matching the exponents of y: 2m = 4

See how we've broken down one big problem into three smaller, more manageable equations? This is a classic strategy in math—divide and conquer! Now, let's solve these equations one by one. The first equation will help us find m, and then we can use that value to find n. It’s like detective work, piecing together clues until we solve the mystery!

Step 4: Solve for m

Let's tackle the first equation we derived from equating coefficients: 2ᵐ = 2². This equation is straightforward, and it's all about understanding what exponents mean. If two powers with the same base are equal, then their exponents must be equal. In simpler terms, if 2 raised to some power m is the same as 2 raised to the power of 2, then m must be 2.

So, we can confidently say that m = 2. Hooray! We've found one of our integers. This is a major breakthrough because now we can use this value of m to solve for n. It’s like finding a key that unlocks the next part of the puzzle. With m in hand, we can move on to the equations involving n and crack this problem wide open.

Step 5: Solve for n

Now that we've nailed down m as 2, let's use that knowledge to find n. We have two equations that involve n: nm = 6 and 2m = 4. However, we've already used 2m = 4 to find m, so let's focus on the equation nm = 6. We know that m = 2, so we can substitute that value into the equation:

n(2) = 6

This simplifies to 2n = 6. To solve for n, we simply divide both sides of the equation by 2:

n = 6 / 2

n = 3

Fantastic! We've found that n = 3. So, by using the value of m we found earlier, we were able to solve for n. This demonstrates how interconnected these equations are and how solving for one variable can lead us to the solution for another. Now that we have both m and n, let's make sure our solution works by plugging them back into the original equation.

Step 6: Verify the Solution

Alright, the moment of truth! We've found that m = 2 and n = 3. To make sure we didn't make any sneaky mistakes along the way, let's plug these values back into the original equation and see if everything checks out:

Original equation: (2xⁿy²)ᵐ = 4x⁶y⁴

Substitute m = 2 and n = 3:

(2x³y²)² = 4x⁶y⁴

Now, let's simplify the left side using the same exponent rules we used earlier:

(2x³y²)² = 2² * (x³)² * (y²)²

= 4 * x^(32) * y^(22)

= 4 * x⁶ * y⁴

Look at that! The left side simplifies to 4x⁶y⁴, which is exactly the same as the right side of the original equation. This confirms that our solution is correct! We've successfully found the values of m and n that make the equation true. High five, guys! We did it!

Conclusion

So, there you have it! We've successfully found the integers m and n that satisfy the equation (2xⁿy²)ᵐ = 4x⁶y⁴. By simplifying the equation, equating coefficients and exponents, and solving the resulting system of equations, we determined that m = 2 and n = 3. We even verified our solution to be absolutely sure. Problems like these are a fantastic way to sharpen our algebra skills and boost our problem-solving confidence. Keep practicing, and you'll become an algebra whiz in no time! Great job, everyone!