Power Problem: Fill In The Blanks & Solve!
Hey guys! Let's dive into a cool math problem where we need to fill in some blanks to make the final answer a power. Sounds interesting, right? This problem involves shelves, notebooks, sheets, and hexagons. We'll break it down step by step so you can totally nail it. Get ready to put on your thinking caps and let's get started!
Understanding the Problem Statement
Okay, so the problem starts like this: "A shelf has 3 shelves, on each shelf there are 4 stacks of 5 notebooks with 6 sheets each, on each sheet there are 2 hexagons. How many...?" Our mission is to figure out what question we can ask at the end of this statement so that the answer can be written as a power. A power, in math terms, is when you multiply a number by itself a certain number of times. Think of it like this: 2 to the power of 3 (written as 2³) is 2 * 2 * 2, which equals 8. We need to tweak the question so that the final number of whatever we're counting can be expressed in this way.
To make this happen, we need to carefully consider each piece of information given. We have shelves, stacks, notebooks, sheets, and hexagons. The key is to multiply these numbers together in a way that results in a number that's a power. This means we need to look for opportunities to create exponents. For example, if we had something like 2 * 2 * 2, we could write it as 2³. So, we're aiming for a similar pattern as we calculate the total number of items. Let's break down the given information step by step to see how we can achieve this.
First, we know there are 3 shelves. Then, on each shelf, there are 4 stacks of notebooks. So, the total number of stacks is 3 * 4. Next, each stack contains 5 notebooks, which adds another multiplication. Each notebook has 6 sheets, and each sheet has 2 hexagons. We’re multiplying all these numbers together: 3 * 4 * 5 * 6 * 2. To get a power, we need to rearrange or adjust these numbers if necessary. This is where the problem gets interesting – we need to think creatively about how to manipulate these values to achieve our goal. Let's move on to the next section to see how we can do just that!
Calculating the Total and Expressing as a Power
Alright, let's get down to the nitty-gritty and calculate the total number of… well, we'll figure out what we're counting soon! So far, we know we have 3 shelves, 4 stacks per shelf, 5 notebooks per stack, 6 sheets per notebook, and 2 hexagons per sheet. If we multiply all these numbers together, we get:
3 * 4 * 5 * 6 * 2 = 720
Okay, 720 is our current total. But can we express 720 as a power? Nope, not easily. That's where the fun part of the problem comes in – we need to tweak the question so that the answer can be expressed as a power. To do this, we need to think about how we can rearrange or modify these numbers. Remember, a power is a number multiplied by itself multiple times (like 2³ = 2 * 2 * 2). We want to find a way to make our final calculation look like that.
Let's break down 720 into its prime factors. Prime factorization means finding the prime numbers that multiply together to give us 720. This will help us see the structure of the number and identify any potential powers hiding within. The prime factorization of 720 is:
720 = 2 * 2 * 2 * 2 * 3 * 3 * 5 = 2⁴ * 3² * 5
Looking at this, we have 2 to the power of 4 (2⁴) and 3 to the power of 2 (3²), which are both powers! However, we also have a 5 hanging out there, which isn't part of a power. This 5 is the key to our puzzle. To make the entire expression a power, we need to somehow get rid of this lone 5 or incorporate it into a power. This is where we need to be a little creative with the original problem statement. We need to figure out what we can ask that will either eliminate the 5 or make it part of a power. Let's explore some options in the next section!
Completing the Statement and Forming the Question
Now comes the creative part – let's complete the statement and form the question so that the answer can be expressed as a power. Remember, we have 720, which breaks down into 2⁴ * 3² * 5. That pesky 5 is preventing us from having a perfect power. So, how can we adjust the problem to get rid of it or incorporate it into a power?
One way to tackle this is to think about what we're counting. Right now, we've calculated the total number of hexagons (3 shelves * 4 stacks * 5 notebooks * 6 sheets * 2 hexagons = 720 hexagons). What if we changed the number of something in the problem to cancel out the 5? For example, if we could somehow divide by 5, we'd be in good shape.
Let's revisit the problem statement. We have:
- 3 shelves
- 4 stacks per shelf
- 5 notebooks per stack
- 6 sheets per notebook
- 2 hexagons per sheet
If we want to divide by 5, we could change the number of notebooks per stack. Instead of 5 notebooks, what if we had a number that, when multiplied by the other factors, would give us a power of 2 and 3? Let's think about this. We currently have 2⁴ * 3² * 5. If we change the number of notebooks per stack to 1 (instead of 5), our new total would be:
3 * 4 * 1 * 6 * 2 = 144
Now, let's break down 144 into its prime factors:
144 = 2 * 2 * 2 * 2 * 3 * 3 = 2⁴ * 3²
Look at that! We have 2⁴ and 3², both powers! So, if we change the number of notebooks per stack to 1, the total number of hexagons becomes 144, which is 2⁴ * 3² = (2² * 3) ² = 12². This means we can express 144 as a power (12 squared). Now, we can complete the statement and form our question:
"A shelf has 3 shelves, on each shelf there are 4 stacks of 1 notebook with 6 sheets each, on each sheet there are 2 hexagons. How many hexagons are there in total?"
The answer is 144, which is 12², a perfect power! We did it! By tweaking the number of notebooks per stack, we were able to make the final answer a power. Let's summarize our solution in the next section.
Solution Summary
Fantastic work, everyone! Let's quickly recap how we solved this problem. Our initial statement was:
"A shelf has 3 shelves, on each shelf there are 4 stacks of 5 notebooks with 6 sheets each, on each sheet there are 2 hexagons. How many...?"
We needed to complete the statement with a question so that the answer could be expressed as a power. We started by multiplying all the numbers together:
3 shelves * 4 stacks * 5 notebooks * 6 sheets * 2 hexagons = 720 hexagons
However, 720 couldn't be directly expressed as a power. So, we broke 720 down into its prime factors:
720 = 2⁴ * 3² * 5
The 5 was the key obstacle. To get rid of it, we decided to change the number of notebooks per stack. Instead of 5 notebooks, we tried 1 notebook. This changed our calculation to:
3 shelves * 4 stacks * 1 notebook * 6 sheets * 2 hexagons = 144 hexagons
Then, we broke down 144 into its prime factors:
144 = 2⁴ * 3²
And voila! 144 can be expressed as a power: 144 = 12².
So, our completed statement and question are:
"A shelf has 3 shelves, on each shelf there are 4 stacks of 1 notebook with 6 sheets each, on each sheet there are 2 hexagons. How many hexagons are there in total?"
The answer is 144 hexagons, which is 12 squared. Great job, guys! You've successfully navigated a tricky problem by breaking it down step by step and thinking creatively. Keep up the awesome work, and remember to always look for the powers hiding in numbers!