Smallest Value Of 2^x * 3^y * 5^z: Math Problem

Hey guys! Today, we're diving into an interesting math problem that involves prime factorization and finding the smallest possible value under specific conditions. It sounds a bit complex, but trust me, we'll break it down step by step. Our main goal is to figure out the smallest value of a number that can be expressed in the form 2^x * 3^y * 5^z, where x, y, and z are distinct positive integers. This means that x, y, and z are all different whole numbers greater than zero. Let's get started and explore how we can solve this! Understanding the concept of prime factorization is crucial here. Remember, prime factorization is the process of breaking down a number into its prime factors – numbers that are only divisible by 1 and themselves. For instance, the prime factors of 12 are 2 and 3 because 12 = 2 * 2 * 3, or 2^2 * 3. In our problem, we're dealing with the prime factors 2, 3, and 5, each raised to a different power (x, y, and z). This makes the problem a bit more intriguing, as we need to consider the impact of these exponents on the final value. The question essentially asks us to minimize the value of the expression 2^x * 3^y * 5^z. To do this, we need to think about how different values of x, y, and z will affect the result. Since 2, 3, and 5 are prime numbers, raising them to different powers will significantly change the outcome. To get the smallest possible value, we should assign the smallest exponents to the largest bases and vice versa. This is because a larger base raised to a smaller power will always be smaller than a smaller base raised to a larger power. For example, 2^3 is smaller than 3^2. So, how do we apply this logic to our problem? Let's find out in the next section!

The Key to Minimizing the Value

To minimize the value of 2^x * 3^y * 5^z, we need to carefully consider which exponents (x, y, and z) should be assigned to which bases (2, 3, and 5). Remember, the exponents must be distinct positive integers. To minimize the overall value, we should assign the smallest possible exponents to the largest bases and the largest exponents to the smallest bases. This is because the impact of a base number increases exponentially as the exponent increases. Think of it this way: raising a larger number to a small power might still result in a relatively small value, but raising a smaller number to a large power can lead to a much larger value. So, let's try to assign the smallest exponents to the largest bases and see what happens. Since x, y, and z must be distinct positive integers, the smallest possible values we can use are 1, 2, and 3. Now, the question is: which of these values should we assign to x, y, and z? Following our strategy of assigning smaller exponents to larger bases, we should assign the smallest exponent (1) to the largest base (5), the next smallest exponent (2) to the next largest base (3), and the largest exponent (3) to the smallest base (2). This gives us the expression 2^3 * 3^2 * 5^1. But why does this assignment work? Let's delve a little deeper into the reasoning behind it. By assigning the exponents in this way, we're ensuring that the larger bases don't grow too quickly. If we were to, say, assign the exponent 3 to the base 5, we would end up with 5^3, which is 125. This is a significantly larger value than 5^1, which is just 5. Similarly, assigning the exponent 2 to the base 3 gives us 3^2 = 9, which is smaller than 3^3 = 27. By keeping the exponents on the larger bases small, we prevent them from dominating the overall value of the expression. Now that we've figured out the optimal assignment of exponents, let's calculate the value and see what we get!

Calculating the Smallest Value

Alright, let's put our plan into action and calculate the smallest possible value of 2^x * 3^y * 5^z. We've already determined that the optimal assignment of exponents is x = 3, y = 2, and z = 1. This means we need to calculate 2^3 * 3^2 * 5^1. Let's break it down step by step. First, let's calculate 2^3. This means 2 multiplied by itself three times: 2 * 2 * 2 = 8. So, 2^3 equals 8. Next, let's calculate 3^2. This means 3 multiplied by itself two times: 3 * 3 = 9. So, 3^2 equals 9. Finally, let's calculate 5^1. Any number raised to the power of 1 is simply the number itself. So, 5^1 equals 5. Now that we have the values of each part of the expression, we can multiply them together: 8 * 9 * 5. Let's do the multiplication. First, we can multiply 8 and 9: 8 * 9 = 72. Then, we multiply 72 by 5: 72 * 5 = 360. So, the smallest possible value of 2^x * 3^y * 5^z is 360. Isn't that neat? We started with a seemingly complex problem and, by breaking it down and using a smart strategy, we were able to find the solution. To recap, we assigned the smallest exponents to the largest bases to minimize the overall value. This gave us the expression 2^3 * 3^2 * 5^1, which equals 360. Now, let's think about why this is the smallest possible value. What if we had assigned the exponents differently? Would we have gotten a smaller result? Let's explore this a bit further in the next section.

Why This is the Smallest Possible Value

We've found that the smallest value of 2^x * 3^y * 5^z is 360 when x = 3, y = 2, and z = 1. But let's really solidify our understanding by considering why this is indeed the smallest possible value. What if we tried a different combination of exponents? Let's explore some alternative scenarios and see how they compare to our result. First, let's think about what would happen if we swapped the exponents of 2 and 5. Instead of 2^3 * 3^2 * 5^1, we would have 2^1 * 3^2 * 5^3. Let's calculate this: 2^1 = 2, 3^2 = 9, and 5^3 = 125. Multiplying these together, we get 2 * 9 * 125 = 2250. Wow, that's significantly larger than 360! This clearly demonstrates that assigning the larger exponent to the larger base (5 in this case) leads to a much bigger result. How about swapping the exponents of 2 and 3? Instead of 2^3 * 3^2 * 5^1, we would have 2^2 * 3^3 * 5^1. Let's calculate this: 2^2 = 4, 3^3 = 27, and 5^1 = 5. Multiplying these together, we get 4 * 27 * 5 = 540. This is also larger than 360, although not as dramatically as the previous example. This further reinforces the idea that assigning smaller exponents to larger bases is the key to minimizing the value. What if we tried assigning the exponent 1 to the base 2? We would have 2^1 * 3^3 * 5^2. Let's calculate this: 2^1 = 2, 3^3 = 27, and 5^2 = 25. Multiplying these together, we get 2 * 27 * 25 = 1350. Again, this is much larger than 360. By trying out these different combinations, we can see a clear pattern: any deviation from our original assignment (x = 3, y = 2, z = 1) results in a larger value. This is because the larger bases (3 and 5) grow much faster when raised to higher powers. By assigning them the smallest possible exponents, we keep their contribution to the overall value relatively small. So, we can confidently conclude that 360 is indeed the smallest possible value of 2^x * 3^y * 5^z when x, y, and z are distinct positive integers. Great job, guys! We've successfully solved this problem and gained a deeper understanding of how exponents and prime factorization work. Keep up the awesome work, and let's tackle more math challenges in the future!