💡First, I need to determine how many roosters Mrs. Fredrickson has. Since 1/4 of the chickens are roosters, I multiply the total number of chickens by 1/4:80 chickens * 1/4 = 20 roostersNext, I'll find out how many hens there are by subtracting the number of roosters from the total number of chickens:80 chickens - 20 roosters = 60 hensNow, I need to calculate how many of these hens lay eggs. Since only three-fourths of the hens lay eggs, I multiply the number of hens by 3/4:60 hens * 3/4 = 45 hens that lay eggsTo find out how many hens do not lay eggs, I subtract the number of hens that lay eggs from the total number of hens:60 hens - 45 hens that lay eggs = 15 hens that do not lay eggsFinally, I add the number of roosters and the number of hens that do not lay eggs to determine the total number of chickens that do not lay eggs:20 roosters + 15 hens that do not lay eggs = 35 chickens that do not lay eggs

💡First, I need to identify the least common multiple (LCM) of the divisors 6, 7, 9, and 18. To do this, I'll start by finding the prime factors of each number:- 6 can be factored into 2 × 3.- 7 is a prime number.- 9 can be factored into 3².- 18 can be factored into 2 × 3².Next, I'll determine the LCM by taking the highest power of each prime number that appears in the factorizations:- The highest power of 2 is 2¹.- The highest power of 3 is 3².- The highest power of 7 is 7¹.Multiplying these together gives the LCM:LCM = 2¹ × 3² × 7¹ = 2 × 9 × 7 = 126.Now, since the number leaves a remainder of 4 when divided by each of these divisors, I'll add the remainder to the LCM:Number = LCM + remainder = 126 + 4 = 130.Therefore, the least number that satisfies the given conditions is 130.

💡Alright, let's tackle this problem step by step. So, we have a geometric sequence {a_n} where the first term a₁ is 1, and the common ratio is q, which is not equal to 1. We need to find the value of k such that a_k equals the product of the first ten terms of the sequence, that is, a₁a₂a₃…a₁₀.First, let me recall what a geometric sequence is. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio, denoted here by q. So, the nth term of the sequence can be written as a_n = a₁ * q^(n-1). Since a₁ is given as 1, this simplifies to a_n = q^(n-1).Now, the problem states that a_k = a₁a₂a₃…a₁₀. Let's write out what each of these terms is in terms of q.a₁ = 1 (given)a₂ = q^(2-1) = qa₃ = q^(3-1) = q²a₄ = q³a₅ = q⁴a₆ = q⁵a₇ = q⁶a₈ = q⁷a₉ = q⁸a₁₀ = q⁹So, the product a₁a₂a₃…a₁₀ would be:1 * q * q² * q³ * q⁴ * q⁵ * q⁶ * q⁷ * q⁸ * q⁹When multiplying terms with the same base, we add their exponents. So, let's add up all the exponents from q⁰ (which is 1) up to q⁹.The exponents are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.To find the sum of these exponents, we can use the formula for the sum of the first n integers, which is n(n + 1)/2. Here, n is 9 because we're summing from 0 to 9, which is 10 terms, but since the first term is 0, it's effectively summing from 1 to 9.Wait, actually, the sum from 0 to 9 is the same as the sum from 1 to 9 plus 0, which doesn't change the sum. So, the sum S is:S = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9Which is the same as S = (9 * 10)/2 = 45.So, the product a₁a₂a₃…a₁₀ = q^45.Now, we have a_k = q^(k - 1) because a_k is the kth term of the geometric sequence, and a_k = a₁ * q^(k - 1) = 1 * q^(k - 1) = q^(k - 1).According to the problem, a_k equals the product of the first ten terms, which we found to be q^45. So, we can set up the equation:q^(k - 1) = q^45Since the bases are the same and q ≠ 1, we can equate the exponents:k - 1 = 45Solving for k:k = 45 + 1k = 46So, the value of k is 46.Let me double-check my steps to ensure I didn't make a mistake.1. I identified that a_n = q^(n - 1) since a₁ = 1.2. I wrote out the first ten terms and expressed their product as q^(0 + 1 + 2 + ... + 9).3. I calculated the sum of exponents from 0 to 9, which is 45.4. I set q^(k - 1) equal to q^45 and solved for k, getting k = 46.Everything seems to check out. I don't see any errors in my reasoning.**Final Answer**boxed{C}

💡Okay, so I'm trying to figure out this problem where I have to determine who is sitting based on some given facts. There are four people involved: M, I, P, and A. Each of them has certain conditions that dictate whether they are sitting or not. Let me try to break this down step by step.First, the problem mentions that M is not sitting. That's straightforward enough. But then, there's this third fact that says if M is not sitting, then I is sitting. Okay, so since M isn't sitting, that means I must be sitting. That makes sense.Now, moving on to the fourth fact, which states that if I is sitting, then P is also sitting. Since we've already established that I is sitting, it follows that P must be sitting as well. So far, so good. I and P are sitting.Next, the second fact tells us that A is not sitting. That's another straightforward piece of information. So, A isn't sitting.Putting it all together: M isn't sitting, A isn't sitting, but I and P are sitting. Does this create any contradictions? Let's see. If M isn't sitting, I is sitting, which in turn means P is sitting. And A isn't sitting. There's no conflict here; all the conditions are satisfied without any overlaps or contradictions.Wait a minute, am I missing something? Let me double-check. The first fact says M isn't sitting, which leads to I sitting via the third fact. I sitting leads to P sitting via the fourth fact. The second fact independently tells us that A isn't sitting. There's no overlap or conflict between these statements. Everything seems to align perfectly.Is there any chance that assuming M isn't sitting could affect other conditions indirectly? For example, could M not sitting somehow influence A's sitting status? The facts don't suggest any direct or indirect relationship between M and A. So, it seems safe to conclude that A isn't sitting based solely on the second fact.Also, are there any hidden assumptions here? I'm assuming that the facts are independent unless stated otherwise. There's no indication that these facts are part of a larger system where one fact affects another beyond what's explicitly stated. So, I think it's safe to take each fact at face value.Another thing to consider is whether the use of "if" in the third and fourth facts implies any kind of dependency or causality that might affect the outcome. For instance, does the fact that I is sitting necessarily mean that P is sitting, or could there be exceptions? Based on the way the facts are presented, it seems like these are strict rules without exceptions. So, if I is sitting, P must be sitting, and there's no room for deviation.Also, the problem mentions that all statements are checked, and no contradictions arose. That gives me more confidence that my interpretation is correct. If there were any hidden contradictions, the problem would likely mention them or hint at the need for further analysis.In summary, starting with M not sitting leads us through the third fact to I sitting, which then leads us through the fourth fact to P sitting. The second fact independently tells us that A isn't sitting. All these pieces fit together without any conflicts or contradictions. Therefore, the feasible seating arrangement is that I and P are sitting, while M and A are not.I think I've covered all the bases here. I've considered each fact individually, checked for any potential overlaps or hidden assumptions, and confirmed that there are no contradictions. This seems like a solid conclusion based on the information provided.

💡Okay, so I've got this equation to solve: the square root of (3 + 2√2)^x plus the square root of (3 - 2√2)^x equals 6. Hmm, that looks a bit intimidating at first glance, but maybe I can break it down step by step.First, let me write it out again to make sure I have it right:√[(3 + 2√2)^x] + √[(3 - 2√2)^x] = 6.Alright, so I have two terms here, both involving square roots of expressions raised to the power of x. The bases inside the square roots are 3 + 2√2 and 3 - 2√2. I notice that these two expressions are conjugates of each other because one has +2√2 and the other has -2√2. That might be useful later on.Let me think about how to simplify this equation. Maybe I can let y be equal to one of these terms to make it easier. Let me set y = √[(3 + 2√2)^x]. Then, what would the other term be?Well, the other term is √[(3 - 2√2)^x]. I wonder if there's a relationship between (3 + 2√2) and (3 - 2√2). Let me calculate their product:(3 + 2√2)(3 - 2√2) = 3^2 - (2√2)^2 = 9 - 8 = 1.Oh, interesting! So, (3 + 2√2) and (3 - 2√2) are reciprocals of each other. That means (3 - 2√2) is equal to 1/(3 + 2√2). So, maybe I can express the second term in terms of y.Let me try that. Since (3 - 2√2) = 1/(3 + 2√2), then (3 - 2√2)^x = [1/(3 + 2√2)]^x = (3 + 2√2)^(-x). Therefore, the square root of that would be √[(3 + 2√2)^(-x)] = (3 + 2√2)^(-x/2).But wait, I defined y as √[(3 + 2√2)^x], which is (3 + 2√2)^(x/2). So, (3 + 2√2)^(-x/2) is just 1/y. So, the second term is 1/y.So, substituting back into the original equation, I have:y + (1/y) = 6.That's a much simpler equation! Now, I can solve for y. Let me write that equation down:y + 1/y = 6.To solve this, I can multiply both sides by y to eliminate the denominator:y^2 + 1 = 6y.Then, bring all terms to one side:y^2 - 6y + 1 = 0.Okay, so now I have a quadratic equation in terms of y. I can use the quadratic formula to solve for y. The quadratic formula is y = [6 ± √(36 - 4*1*1)] / 2.Calculating the discriminant first: √(36 - 4) = √32. Simplifying √32, that's √(16*2) = 4√2.So, plugging that back into the formula, we get:y = [6 ± 4√2] / 2.Simplifying that, divide numerator and denominator by 2:y = 3 ± 2√2.So, y can be either 3 + 2√2 or 3 - 2√2. Now, I need to relate this back to x.Remember, y was defined as √[(3 + 2√2)^x], which is equal to (3 + 2√2)^(x/2). So, let's set that equal to each of the solutions for y.First, if y = 3 + 2√2, then:(3 + 2√2)^(x/2) = 3 + 2√2.Since the bases are the same, I can set the exponents equal to each other:x/2 = 1.Multiplying both sides by 2:x = 2.Okay, that's one solution. Now, let's consider the other case where y = 3 - 2√2.So, (3 + 2√2)^(x/2) = 3 - 2√2.Hmm, 3 - 2√2 is actually the reciprocal of 3 + 2√2, right? Because earlier, we saw that (3 + 2√2)(3 - 2√2) = 1. So, 3 - 2√2 = 1/(3 + 2√2).Therefore, 3 - 2√2 can be written as (3 + 2√2)^(-1). So, substituting back in:(3 + 2√2)^(x/2) = (3 + 2√2)^(-1).Again, since the bases are the same, set the exponents equal:x/2 = -1.Multiplying both sides by 2:x = -2.So, the two solutions are x = 2 and x = -2.Wait, let me double-check these solutions to make sure they work in the original equation.First, let's test x = 2:√[(3 + 2√2)^2] + √[(3 - 2√2)^2].Calculating each term:√[(3 + 2√2)^2] = 3 + 2√2.Similarly, √[(3 - 2√2)^2] = 3 - 2√2.Adding them together: (3 + 2√2) + (3 - 2√2) = 6. That works.Now, let's test x = -2:√[(3 + 2√2)^(-2)] + √[(3 - 2√2)^(-2)].Simplify each term:√[(3 + 2√2)^(-2)] = 1 / √[(3 + 2√2)^2] = 1 / (3 + 2√2).Similarly, √[(3 - 2√2)^(-2)] = 1 / √[(3 - 2√2)^2] = 1 / (3 - 2√2).So, adding them together:1/(3 + 2√2) + 1/(3 - 2√2).To add these, I need a common denominator. Let's rationalize each denominator:1/(3 + 2√2) = (3 - 2√2)/[(3 + 2√2)(3 - 2√2)] = (3 - 2√2)/1 = 3 - 2√2.Similarly, 1/(3 - 2√2) = (3 + 2√2)/[(3 - 2√2)(3 + 2√2)] = (3 + 2√2)/1 = 3 + 2√2.Adding these together: (3 - 2√2) + (3 + 2√2) = 6. That also works.So, both solutions satisfy the original equation. Therefore, the solutions are x = 2 and x = -2.I think that's all. I don't see any mistakes in my reasoning, and both solutions check out when plugged back into the original equation. So, I'm confident that these are the correct solutions.**Final Answer**The solutions are boxed{2} and boxed{-2}.**Final Answer**The solutions are boxed{2} and boxed{-2}.

💡To find the absolute value of (-frac{1}{3}), I start by recalling the definition of absolute value, which represents the distance of a number from 0 on the number line, irrespective of its direction.Given the number (-frac{1}{3}), its absolute value is calculated by removing the negative sign, resulting in (frac{1}{3}).This means that the distance of (-frac{1}{3}) from 0 on the number line is (frac{1}{3}), without considering whether the original number was positive or negative.Therefore, the correct answer is (boxed{B}).