💡Alright, so I have this logarithmic equation to solve: log base 7 of (log base 5 of (log base 2 of x)) equals 0. Hmm, that's a bit of a mouthful, but let's break it down step by step. First, I remember that if log base a of b equals c, then b equals a to the power of c. So, applying that to the outermost logarithm here, which is log base 7 of something equals 0, that something must be 7 to the power of 0, right? And anything to the power of 0 is 1. So, that simplifies the equation a bit.So, now I have log base 5 of (log base 2 of x) equals 1. Okay, same idea applies here. If log base 5 of something equals 1, then that something must be 5 to the power of 1, which is just 5. So, now I know that log base 2 of x equals 5.Alright, moving on to the innermost logarithm. If log base 2 of x equals 5, then x must be 2 to the power of 5. Calculating that, 2 to the 5th power is 32. So, x equals 32.But wait, the question isn't just asking for x; it's asking for x raised to the power of -1/3. So, I need to compute 32 to the power of -1/3. Negative exponents mean reciprocals, so 32 to the -1/3 is the same as 1 divided by 32 to the 1/3.Now, 32 to the 1/3 is the cube root of 32. Hmm, 32 isn't a perfect cube, but I can express it in terms of smaller numbers. I know that 32 is 2 to the 5th power, so 32 is 2^5. Therefore, the cube root of 32 is the cube root of 2^5, which can be written as 2^(5/3). Breaking that down further, 2^(5/3) is the same as 2^(1 + 2/3), which is 2^1 times 2^(2/3). So, that's 2 times the cube root of 2 squared. The cube root of 2 squared is the cube root of 4. So, putting it all together, 32^(1/3) is 2 times the cube root of 4.Therefore, 32^(-1/3) is 1 divided by (2 times the cube root of 4). So, that simplifies to 1/(2 * cube root of 4). Looking back at the answer choices, that's option C: 1/(2 * cube root of 4).Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from the outside logarithm, log base 7 of something equals 0, so that something is 1. Then log base 5 of (log base 2 of x) equals 1, so log base 2 of x is 5, making x 32. Then x^(-1/3) is 32^(-1/3), which is 1 over the cube root of 32. Breaking down 32 into 2^5, the cube root of 2^5 is 2^(5/3), which is 2 * 2^(2/3) or 2 * cube root of 4. So, 1 over that is 1/(2 * cube root of 4). Yep, that seems right.I think I'm confident with that answer. It's option C.**Final Answer**boxed{C}

💡Okay, so I have this problem where the terminal side of angle α passes through the point P(-4, 3). I need to find sin θ, cos θ, and tan θ for part (1), and then evaluate this trigonometric expression for part (2). Hmm, let me start with part (1).First, I remember that when dealing with angles in standard position, the terminal side passing through a point (x, y) means that we can use the coordinates to find the trigonometric functions. I think we need to find the distance from the origin to the point P(-4, 3) because that will be the radius r in the unit circle concept. So, the distance formula is r = sqrt(x² + y²). Plugging in the values, x is -4 and y is 3. So, r = sqrt((-4)² + 3²) = sqrt(16 + 9) = sqrt(25) = 5. Okay, so r is 5. That makes sense.Now, for sin θ, I recall that it's y over r. So, sin θ = y/r = 3/5. That seems straightforward. Next, cos θ is x over r. So, cos θ = x/r = -4/5. Wait, is that right? Since x is negative, cos θ will be negative. Yeah, that makes sense because the point is in the second quadrant where cosine is negative and sine is positive.Then, tan θ is y over x. So, tan θ = y/x = 3/(-4) = -3/4. Okay, so tan θ is negative because sine is positive and cosine is negative in the second quadrant, so tangent should be negative. That checks out.So, for part (1), I have sin θ = 3/5, cos θ = -4/5, and tan θ = -3/4. I think that's it for part (1).Moving on to part (2), the expression is [cos(θ - π/2) / sin(π/2 + θ)] * sin(θ + π) * cos(2π - θ). Hmm, that looks a bit complicated, but maybe I can simplify it using trigonometric identities.Let me recall some identities. I remember that cos(θ - π/2) is equal to sin θ because cosine of an angle minus π/2 is sine of that angle. Similarly, sin(π/2 + θ) is equal to cos θ because sine of π/2 plus an angle is cosine of that angle. So, substituting these in, the first part becomes [sin θ / cos θ].Then, sin(θ + π) is equal to -sin θ because sine has a period of 2π and sin(θ + π) is like shifting the angle by π, which flips the sign. So, sin(θ + π) = -sin θ.Lastly, cos(2π - θ) is equal to cos θ because cosine is an even function, and subtracting θ from 2π doesn't change the value; it's like reflecting over the x-axis, but cosine is positive in the fourth quadrant as well. So, cos(2π - θ) = cos θ.Putting it all together, the expression becomes [sin θ / cos θ] * (-sin θ) * cos θ. Let me write that out:[sin θ / cos θ] * (-sin θ) * cos θ.Now, I can simplify this step by step. First, [sin θ / cos θ] is tan θ, but maybe it's better to just multiply it out. Let's see:Multiply [sin θ / cos θ] by (-sin θ): that gives (-sin² θ / cos θ).Then, multiply that by cos θ: (-sin² θ / cos θ) * cos θ = -sin² θ.Because the cos θ in the denominator and the cos θ in the numerator cancel each other out. So, we're left with -sin² θ.Now, from part (1), we know sin θ is 3/5. So, sin² θ is (3/5)² = 9/25. Therefore, -sin² θ is -9/25.So, the value of the entire expression is -9/25.Wait, let me double-check to make sure I didn't make any mistakes. Starting from the beginning:1. Found r correctly as 5.2. Calculated sin θ = 3/5, cos θ = -4/5, tan θ = -3/4. All correct.3. For part (2), used identities: - cos(θ - π/2) = sin θ - sin(π/2 + θ) = cos θ - sin(θ + π) = -sin θ - cos(2π - θ) = cos θ4. Substituted these into the expression and simplified step by step, ending up with -sin² θ.5. Plugged in sin θ = 3/5, squared it, and applied the negative sign.Everything seems to check out. I don't think I made any calculation errors. So, I'm confident that the final answer is -9/25.

💡Okay, so I have this math problem here: I need to figure out which of the given options is closest to the value of 503 divided by 0.251. The options are 2000, 1500, 200, 2500, and 3000. Hmm, let me think about how to approach this.First, I know that dividing by a decimal can sometimes be tricky, so maybe I can simplify the numbers to make the calculation easier. Let's see, 503 is pretty close to 500, right? And 0.251 is almost 0.25. So if I approximate both numbers, I can maybe get a rough estimate.So, if I take 500 divided by 0.25, what does that give me? Well, dividing by 0.25 is the same as multiplying by 4 because 0.25 is a quarter, and there are four quarters in a whole. So, 500 times 4 is 2000. That seems straightforward.But wait, I approximated both numbers. The original numerator was 503, which is a bit more than 500, and the denominator was 0.251, which is slightly more than 0.25. So, does that mean my estimate of 2000 is a little low or a little high?Let me think. If both the numerator and denominator were slightly larger than my approximations, how does that affect the result? Well, increasing the numerator would make the result larger, and increasing the denominator would make the result smaller. So, these two effects kind of cancel each other out a bit. Maybe my estimate of 2000 is pretty close.But just to be sure, maybe I should do a more precise calculation. Let's see, 503 divided by 0.251. Hmm, I can write this as 503 multiplied by (1 divided by 0.251). Let me calculate 1 divided by 0.251 first.Calculating 1 divided by 0.251... Well, 0.251 times 4 is 1.004, which is just a bit more than 1. So, 1 divided by 0.251 is approximately 3.984. That's almost 4, but a tiny bit less.So, if I multiply 503 by 3.984, I should get a more accurate estimate. Let's do that step by step. First, 500 times 3.984 is 1992. Then, 3 times 3.984 is about 11.952. Adding those together, 1992 plus 11.952 gives me approximately 2003.952.Wow, that's really close to 2000. So, my initial estimate was pretty accurate. Therefore, 503 divided by 0.251 is approximately 2004, which is very close to 2000.Looking back at the options, 2000 is option A, and the other options are 1500, 200, 2500, and 3000. Clearly, 2000 is the closest to 2004. So, I think the answer is A, 2000.But just to double-check, maybe I should consider the exact division without approximating. Let's see, 503 divided by 0.251. I can write this as 503 multiplied by (1000/251) because 0.251 is 251/1000. So, 503 times 1000 is 503,000, and then divided by 251.Calculating 503,000 divided by 251. Hmm, let's see. 251 times 2000 is 502,000. So, 503,000 minus 502,000 is 1,000. So, 251 goes into 1,000 about 3 times because 251 times 3 is 753. So, that leaves a remainder of 247. So, the exact value is 2003 and 247/251, which is approximately 2003.984.So, that confirms my earlier calculation. The exact value is about 2003.984, which is just a bit over 2000. So, among the options given, 2000 is definitely the closest.I think I'm confident now that the correct answer is A, 2000.

💡First, I need to understand the structure of the figure made with forty toothpicks. It mentions that there's an additional row on top and bottom compared to the previous figure. This suggests that the figure is a larger, symmetric arrangement of toothpicks forming triangles.Next, I'll consider the types of triangles present. There are upward-pointing and downward-pointing triangles. Each triangle is formed by three toothpicks. The additional rows likely increase the number of both upward and downward-pointing triangles.To eliminate all triangles, I need to remove toothpicks in such a way that no three toothpicks can form a triangle. A strategic approach would be to remove all horizontal toothpicks because every triangle requires at least one horizontal toothpick to complete its base.Counting the horizontal toothpicks in each row:- Top row: 5 horizontal toothpicks- Second row: 10 horizontal toothpicks- Third row: 10 horizontal toothpicks- Fourth row: 10 horizontal toothpicks- Bottom row: 5 horizontal toothpicksAdding these together gives a total of 40 horizontal toothpicks. Removing all of these ensures that no triangles can be formed, as the base of every potential triangle is eliminated.Therefore, removing all 40 horizontal toothpicks is necessary to ensure no triangles remain in the figure.

💡Okay, so I've got this math problem here: "There are 20 houses in a block. If a mailman has to give 32 pieces of junk mail to each house in each block. How many pieces of junk mail should he give each block?" Hmm, let's see. I need to figure out how many pieces of junk mail the mailman should give to each block.Alright, first, I need to understand what the problem is asking. It says there are 20 houses in a block, and the mailman has to give 32 pieces of junk mail to each house. So, for each house, he gives out 32 pieces. Since there are 20 houses, I guess I need to find out the total number of pieces he gives out in one block.Wait, so if each house gets 32 pieces, and there are 20 houses, then the total should be 32 multiplied by 20, right? Let me write that down: 32 times 20. Hmm, 32 times 20. Okay, 32 times 2 is 64, so 32 times 20 should be 640. So, the mailman should give out 640 pieces of junk mail to each block.But let me double-check that to make sure I didn't make a mistake. If there are 20 houses and each gets 32 pieces, then adding 32 twenty times should give the same result. So, 32 plus 32 is 64, plus another 32 is 96, and so on. But that would take a long time. Maybe there's a quicker way.Oh, right, multiplication is just repeated addition, so 32 times 20 is the same as adding 32 twenty times. So, 32 times 20 is indeed 640. Okay, that makes sense.Wait, but the problem says "each block." Does that mean there's more than one block? Or is it just asking about one block? The way it's phrased, it seems like it's asking about each individual block. So, if there are multiple blocks, each block would receive 640 pieces. But since the question specifically asks "how many pieces of junk mail should he give each block," I think it's referring to one block.So, to summarize, if there are 20 houses in a block and each house gets 32 pieces of junk mail, then the total number of pieces per block is 32 multiplied by 20, which equals 640 pieces.I think that's it. I don't see any other factors or details in the problem that I need to consider. It seems straightforward: multiply the number of houses by the number of pieces per house to get the total per block.

💡First, I need to determine the original cost price of the article, which I'll denote as ( C ).The man sells the article at a 25% profit, so the original selling price ( S ) can be expressed as:[ S = C + 0.25C = 1.25C ]If the cost price were reduced by 20%, the new cost price ( C_{text{new}} ) would be:[ C_{text{new}} = C - 0.20C = 0.80C ]If the selling price were reduced by Rs. 16.80, the new selling price ( S_{text{new}} ) would be:[ S_{text{new}} = S - 16.80 = 1.25C - 16.80 ]With the new cost price and a 30% profit, the new selling price can also be expressed as:[ S_{text{new}} = C_{text{new}} + 0.30C_{text{new}} = 0.80C + 0.24C = 1.04C ]Setting the two expressions for ( S_{text{new}} ) equal to each other:[ 1.25C - 16.80 = 1.04C ]Solving for ( C ):[ 0.21C = 16.80 ][ C = frac{16.80}{0.21} = 80 ]Therefore, the original cost price of the article is Rs. 80.