Proving A Number Is A Perfect Square: A Math Breakdown

Hey guys! Let's dive into a cool math problem that's all about proving that a number is a perfect square. We're going to break down how to show that the number a = (3^21 + 3^20 + 3^19) : 39 is indeed a perfect square. This problem combines exponents, factoring, and division – a great way to flex those math muscles! We will break it down step by step to ensure everyone understands the concept and the process. The main goal here is to transform the given expression into a form that clearly reveals it as the square of an integer. This involves careful manipulation of exponents, clever use of factoring techniques, and a solid grasp of arithmetic operations. This might seem a little intimidating at first, but trust me, with each step, it'll become clearer and clearer. Think of it like a puzzle; we're just rearranging the pieces until the solution clicks into place. So, grab your pencils, and let's get started. We'll start by looking at the given number and figure out the best way to simplify it so that we can clearly see the pattern. Get ready to have your mind blown!

Firstly, we have to simplify the number by using exponent rules. The number is a = (3^21 + 3^20 + 3^19) : 39. When we see exponents, our first instinct should be to look for common factors. Notice that each term in the parenthesis has a power of 3. Moreover, the smallest power is 19. That gives us a significant clue, so we need to factor out the smallest power of 3. We are going to factor out 3^19, this is a super important step. Remember, factoring is like the reverse of the distributive property. It lets us rewrite an expression in a more convenient form, making it easier to work with. Once we factor, we are going to look for other simplifications. This could be combining like terms, canceling out terms, or simplifying fractions. The key is to keep an eye out for opportunities to make the expression simpler and more manageable. The trick here is to rewrite each term so it has 3^19 as a factor. Now, we are going to factor out 3^19 from the numerator to make it easier to work with. Let's do it and see what happens! When we factor out 3^19, the expression becomes a = [3^19 * (3^2 + 3^1 + 1)] : 39. See, how factoring simplifies things? Now the expression inside the parenthesis is much easier to work with. It's much simpler now, right? So, this transformation is really key to seeing how this number is a perfect square. Now that we have factored out 3^19, we are closer to our goal!

Simplifying the Expression Step-by-Step

Alright, let's keep going. We've got our expression factored, now it's time to simplify it. So, let's look at the part inside the brackets. We need to do some calculations. Inside the parentheses, we have 3^2 + 3^1 + 1. Calculating this gives us 9 + 3 + 1 = 13. Now our expression looks like this: a = [3^19 * 13] : 39. Remember, our goal is to show that a is a perfect square. What do you think we should do next? We need to simplify the division by 39. To make this easier, let's break down the denominator. Now, 39 can be written as 3 * 13. This helps us see how we can simplify the expression further. Watch this! We can rewrite the expression as a = (3^19 * 13) : (3 * 13). Now, we can cancel out the common factors, right? When we divide, we can cancel out the 13 in the numerator and the denominator. We can also simplify by dividing 3^19 by 3. This is using the exponent rules and simplifying the fraction. This is where the magic starts to happen! When we cancel out the 13 and simplify the powers of 3, we get a = 3^18. Isn't that awesome?

So now we have simplified the original expression into a much simpler form. The simplification process is not always straightforward, but the basic idea is always the same: break down the problem into smaller, more manageable steps. By factoring, simplifying, and using exponent rules, we have made significant progress in simplifying the given expression. Now that we have 3^18, we can work towards our goal. We want to show that a is a perfect square. Now that we have a simplified form, what should we do to get closer to the final solution? We should rewrite the expression to reveal its perfect square form. This is the crucial step. It's where we demonstrate that the given number is indeed the square of an integer. By carefully manipulating the exponents and applying the rules of algebra, we can transform the expression into a form that clearly shows its perfect square nature. Now, this is where we have to remember the definition of the perfect square. A number is a perfect square if it is the product of an integer multiplied by itself. Now that we have the simplified form a = 3^18, we can show that it's a perfect square. Remember that a perfect square is a number that can be expressed as the square of another integer. So, we're looking to rewrite 3^18 in the form of (something)^2. The goal is to express the result as an integer squared, demonstrating it is a perfect square.

Showing that 'a' is a Perfect Square

Okay, guys, here comes the final stretch! We're almost there. Now that we've simplified to a = 3^18, we can rewrite it to prove that a is a perfect square. Remember that a perfect square is a number that can be written as (something)^2. We can rewrite 3^18 as (3^9)^2. See that? We've successfully expressed a as the square of an integer (3^9)! Since 3^9 is an integer, and a can be written as the square of this integer, we can officially declare that a is a perfect square. We've proven it! That's it, we are done! We started with a complex expression and, through careful simplification and understanding of the properties of exponents and perfect squares, successfully demonstrated that a is a perfect square. By breaking down the problem into smaller, manageable steps, we were able to systematically simplify and rewrite the expression. This highlights the importance of not only knowing the rules of mathematics but also how to apply them strategically. Remember, in math, often the path to the solution is not obvious, and it requires a combination of knowledge, practice, and a bit of creativity. So, we started with a = (3^21 + 3^20 + 3^19) : 39 and ended up showing that a = (3^9)^2. Now, wasn't that a great math adventure?

Therefore, the number a = (3^21 + 3^20 + 3^19) : 39 is a perfect square, because it can be expressed as (3^9)^2. We used factoring, exponent rules, and division to simplify the expression and prove our point. This is a classic example of how understanding mathematical concepts can help us solve complex problems. By breaking down the problem into smaller steps and using the right tools, we were able to reach our goal. Congratulations, everyone! We successfully proved that a is a perfect square. Remember, with practice, you'll get better and better at these kinds of problems.

In Conclusion: We've shown that a = (3^21 + 3^20 + 3^19) : 39 is a perfect square. We did this by factoring, simplifying, and rewriting the expression to (3^9)^2. This proves that a is indeed a perfect square. Awesome work, everyone! Keep practicing, and you'll become math wizards in no time.