💡First, identify the total number of ways to choose two distinct numbers from the set ({2, 3, 8, 9}). Since the order matters (i.e., (a) and (b) are distinct), there are (4 times 3 = 12) possible ordered pairs.Next, determine which of these pairs satisfy the condition that (log_{a}b) is an integer. This means that (b) must be a power of (a).- For (a = 2): - (b = 8) because (2^3 = 8).- For (a = 3): - (b = 9) because (3^2 = 9).Thus, there are 2 valid pairs: ((2, 8)) and ((3, 9)).Finally, calculate the probability by dividing the number of valid pairs by the total number of pairs:[text{Probability} = frac{2}{12} = frac{1}{6}]

💡Okay, so I have this problem about systematic sampling, and I need to figure out which number between 33 and 48 should be selected if the number 7 is drawn. Let me try to break this down step by step.First, I remember that systematic sampling involves selecting every k-th individual from a population. In this case, there are 800 students, and they want to select 50 students. So, the first thing I think of is calculating the sampling interval, which is usually the total population divided by the sample size. Let me write that down:Sampling interval (k) = Total population / Sample size = 800 / 50 = 16.Okay, so every 16th student will be selected. Now, the students are numbered from 1 to 800, and the number 7 was drawn. I guess that means 7 is the starting point. So, the first selected student is number 7.But wait, the question is asking about a group of 16 numbers between 33 and 48. That seems like a specific range. I need to figure out which number in this range should be selected based on the systematic sampling method.Let me think. If the sampling interval is 16, then after selecting 7, the next selected number would be 7 + 16 = 23, right? Then the one after that would be 23 + 16 = 39. Hmm, 39 is within the range of 33 to 48. So, is 39 the number that should be selected from that group?But wait, let me make sure I'm doing this correctly. Sometimes in systematic sampling, especially when the starting point is less than the sampling interval, you might have to adjust for wrapping around or something. But in this case, since 7 is less than 16, adding 16 each time should work fine.Let me list out the selected numbers to see the pattern:1st: 72nd: 7 + 16 = 233rd: 23 + 16 = 394th: 39 + 16 = 55And so on.So, the third selected number is 39, which falls within the range of 33 to 48. That seems to match one of the options given: B: 39.But just to double-check, let me think about the groups. The students are numbered from 1 to 800, and the sampling interval is 16. So, the first group would be 1-16, the second group 17-32, the third group 33-48, and so on.Since the starting point is 7, which is in the first group (1-16), the next group would be the second group (17-32), and the third group is 33-48. So, the third selected number should be in the third group, which is 33-48.Calculating it again: 7 + 2*16 = 7 + 32 = 39. Yep, that's within 33-48.So, I think the correct answer is 39, which is option B.Just to make sure I'm not missing anything, let me consider if there's another way to interpret the problem. Maybe the 16 numbers between 33 and 48 are considered as a block, and I need to find which number in that block corresponds to the starting point 7.If I think of it that way, the block from 33-48 is the third block (since 1-16 is first, 17-32 is second, 33-48 is third). So, the starting point in the third block would be 7 + 2*16 = 39, which is the same as before.Alternatively, if I use modular arithmetic, 7 mod 16 is 7, so adding that to the start of the third block (33) would give 33 + 7 = 40. But wait, that's different from before. So, which one is correct?Hmm, I think the confusion arises from whether we're adding the interval or using modular arithmetic. In systematic sampling, once you have the starting point, you keep adding the interval. So, starting at 7, adding 16 gives 23, then adding another 16 gives 39. So, 39 is the correct number in the third block.The modular approach might be used in circular systematic sampling, where you wrap around if you exceed the population size, but in this case, since 39 is within 1-800, we don't need to worry about wrapping around.Therefore, I'm confident that 39 is the correct answer.

💡Alright, so I have this expression to factorize: 8a squared minus 2. Hmm, okay, let's see. I remember that factorizing means breaking down an expression into simpler parts that can be multiplied together to get the original expression. So, I need to find two or more expressions that when multiplied give me 8a² - 2.First, I notice that both terms in the expression have a common factor. The first term is 8a² and the second is -2. The greatest common factor (GCF) of 8 and 2 is 2. So, I can factor out a 2 from both terms. Let me write that down:8a² - 2 = 2(4a² - 1)Okay, so now I have 2 times (4a² - 1). Now, I need to see if I can factorize 4a² - 1 further. Hmm, 4a² is a perfect square because it's (2a) squared, and 1 is also a perfect square because it's 1 squared. So, 4a² - 1 is a difference of squares.I remember that the difference of squares formula is a² - b² = (a + b)(a - b). Applying this to 4a² - 1, where a is 2a and b is 1, I get:4a² - 1 = (2a + 1)(2a - 1)So, putting it all together, the original expression becomes:8a² - 2 = 2(2a + 1)(2a - 1)Let me double-check to make sure this is correct. If I multiply 2 by (2a + 1) and then by (2a - 1), I should get back to the original expression.First, multiply (2a + 1)(2a - 1):(2a + 1)(2a - 1) = (2a)(2a) + (2a)(-1) + (1)(2a) + (1)(-1)= 4a² - 2a + 2a - 1= 4a² - 1Okay, that checks out. Now, multiply by 2:2(4a² - 1) = 8a² - 2Yes, that's the original expression. So, the factorization is correct.I think that's it. I've factored out the greatest common factor first and then applied the difference of squares formula to the remaining expression. It all seems to make sense and checks out when I multiply it back together. So, the final factorized form of 8a² - 2 is 2 times (2a + 1) times (2a - 1).

💡First, I need to find the value of ( h(2) ) by substituting ( x = 2 ) into the function ( h(x) = 4x^2 - 8 ).[h(2) = 4(2)^2 - 8 = 16 - 8 = 8]Next, I use the result from ( h(2) ) to find ( h(h(2)) = h(8) ) by substituting ( x = 8 ) into the function.[h(8) = 4(8)^2 - 8 = 4 cdot 64 - 8 = 256 - 8 = 248]Therefore, the value of ( h(h(2)) ) is (boxed{248}).

💡First, I need to determine the value of ( y ) such that the distance between the points ( (3, 4) ) and ( (-5, y) ) is 12 units.I'll use the distance formula:[sqrt{(3 - (-5))^2 + (y - 4)^2} = 12]Simplifying the expression inside the square root:[sqrt{(3 + 5)^2 + (y - 4)^2} = 12 sqrt{8^2 + (y - 4)^2} = 12 sqrt{64 + (y - 4)^2} = 12]Next, I'll square both sides to eliminate the square root:[64 + (y - 4)^2 = 144]Subtracting 64 from both sides:[(y - 4)^2 = 80]Taking the square root of both sides:[y - 4 = pm sqrt{80} y - 4 = pm 4sqrt{5}]Finally, solving for ( y ):[y = 4 pm 4sqrt{5}]Since ( y > 0 ), both solutions are valid. However, typically the positive value is considered:[y = 4 + 4sqrt{5}]

💡First, I need to determine the number of gaps between the 18 trees. Since there is one tree at each end of the yard, the number of gaps is one less than the number of trees.Next, I'll calculate the total length of the yard by multiplying the number of gaps by the distance between two consecutive trees, which is 15 meters.Finally, I'll present the total length of the yard in a clear and concise manner.