Solving For 'a' In A Square Divided Into Smaller Squares

Hey guys, let's dive into a fun geometry problem! We've got a square, ABCD, and it's been cleverly divided into 16 identical smaller squares. We're given some key information: the side length of the big square, |AB|, is equal to 64 raised to the power of 'a', and the area of one of the little squares is 16 raised to the power of (a + 1). Our mission? To find the value of the real number 'a'. Sounds exciting, right? Let's break it down step by step to find the value of a. This problem is a fantastic blend of geometry and algebra, perfect for sharpening our problem-solving skills. So, grab your pencils and let's get started!

Unpacking the Problem

Alright, first things first, let's clarify what we know. We have a square ABCD. Because it's a square, all four sides are equal in length. We are also given that the length of side AB is 64^a. Since all sides of a square are equal, we also know that BC, CD, and DA are all equal to 64^a. This is a very important piece of the puzzle, because it gives us a starting point to work with.

Next, the big square is divided into 16 smaller squares. Think of it like a grid, 4 squares across and 4 squares down. This means the side length of each small square is a fraction of the side length of the large square. In fact, since the big square has been divided into 16 equal smaller squares, this means that the side of one of the smaller squares is one-fourth the side of the larger square. Since we know the side length of the big square, this lets us calculate the side length of the smaller square. The area of one of the smaller squares is given as 16^(a+1). The area of a square is calculated by multiplying the side length by itself, or side length squared. Using this knowledge, we can derive an equation to solve for a.

So, we have two primary pieces of information. Firstly, the side length of the larger square is expressed as an exponential function. And secondly, the area of one of the smaller squares is given as another exponential function. These pieces of information allow us to set up an equation that we can solve for a.

Setting Up the Equations

Now, let's get mathematical, shall we? We know that the side length of the larger square, |AB|, is 64^a. Because the square is divided into 16 smaller squares, and these are all equal, we can determine the side length of one of the smaller squares. Imagine the big square split into 4 rows and 4 columns of smaller squares. This means the side length of a small square is one-fourth the side length of the large square. Therefore, the side length of a small square is (64^a) / 4.

Next, we know the area of one small square is 16^(a+1). The area of a square is its side length squared. So, if we square the side length of a small square, we should get its area. Therefore, we can write the equation: [(64^a) / 4]^2 = 16^(a+1). This equation brings together our geometric knowledge and our algebraic understanding. This single equation is the key to unlocking the value of a. It's where geometry and algebra meet!

Notice that both sides of this equation are expressed using exponents. Our next step is to simplify the equation, combining like terms, and finding a common base to make the solution easier. Before we go any further, take a moment to look at the equation and try to figure out the next step. Practice makes perfect, and this is a great opportunity to practice your problem-solving skills.

Solving for 'a'

Alright, let's get down to business and solve for 'a'. We have the equation: [(64^a) / 4]^2 = 16^(a+1). Our goal is to isolate a and find its value. First, let's simplify the left side of the equation. (64^a / 4)^2 is equivalent to (64^(2a)) / 16. This is because when you square a fraction, you square both the numerator and the denominator. So our equation now looks like this: 64^(2a) / 16 = 16^(a+1).

To make things easier, let's express all the numbers as powers of 2. We know that 64 is 2^6 and 16 is 2^4. Also, notice that 16 is present in the numerator and denominator, which means we can cancel out the values. Let's rewrite the equation: (2^(6 * 2a)) / 2^4 = 2^(4 * (a+1)). This simplifies to: 2^(12a) / 2^4 = 2^(4a + 4). Further, we can rewrite the left side of the equation as 2^(12a - 4), due to the properties of exponents where dividing exponents with the same base means subtracting the powers. Now our equation looks like this: 2^(12a - 4) = 2^(4a + 4). Since the bases are the same, we can equate the exponents: 12a - 4 = 4a + 4.

Now we're down to a simple linear equation! Let's solve it. Subtract 4a from both sides: 8a - 4 = 4. Add 4 to both sides: 8a = 8. Finally, divide both sides by 8: a = 1. Congratulations, we've found the value of a!

Verification and Conclusion

We found that a equals 1. But, as good mathematicians, we should always double-check our work. Let's plug a = 1 back into our original equation and see if it holds true. Remember, our original equation was: [(64^a) / 4]^2 = 16^(a+1).

If a = 1, then [(64^1) / 4]^2 = 16^(1+1). Simplifying, we get (64 / 4)^2 = 16^2. Which is 16^2 = 16^2, which is 256 = 256. And there you have it, our answer checks out!

So, the real number a that satisfies the conditions of this problem is 1. This problem demonstrates how geometric relationships and algebraic techniques can be combined to solve complex problems. It requires a solid understanding of squares, areas, and the properties of exponents. Well done, guys! You've successfully navigated this geometric puzzle. Keep practicing, and you'll become even better at these types of problems. Remember to always double-check your work, and don't be afraid to break down a problem into smaller, more manageable steps. Maths is fun and exciting, isn't it? Keep up the good work! And that's all, folks!