💡First, I need to solve the equation ((1 + i)z = 1 - z) for the complex number (z).I'll start by bringing all terms involving (z) to one side of the equation:[(1 + i)z + z = 1]This simplifies to:[z(1 + i + 1) = 1][z(2 + i) = 1]Next, I'll solve for (z) by dividing both sides by (2 + i):[z = frac{1}{2 + i}]To simplify this expression, I'll multiply the numerator and the denominator by the complex conjugate of the denominator, which is (2 - i):[z = frac{1 times (2 - i)}{(2 + i)(2 - i)} = frac{2 - i}{4 + 1} = frac{2 - i}{5} = frac{2}{5} - frac{1}{5}i]From this, I can see that the imaginary part of (z) is (-frac{1}{5}).

💡First, identify the given information: the width of the garden is 13 meters, and the length is three times the width.Calculate the length by multiplying the width by 3:Length = 3 × 13 meters = 39 meters.Next, use the formula for the area of a rectangle:Area = Length × Width.Substitute the known values into the formula:Area = 39 meters × 13 meters = 507 square meters.Finally, present the calculated area as the solution.

💡First, I need to determine ( f^{-1}(6) ). From the table, I see that ( f(2) = 6 ), so ( f^{-1}(6) = 2 ).Next, I need to find ( f^{-1}(2) ). Looking at the table again, ( f(3) = 2 ), so ( f^{-1}(2) = 3 ).Finally, I need to determine ( f^{-1}(3) ). From the table, ( f(5) = 3 ), so ( f^{-1}(3) = 5 ).Therefore, ( f^{-1}(f^{-1}(f^{-1}(6))) = 5 ).

💡To determine the number of ways to distribute 5 indistinguishable balls into 3 distinguishable boxes, we can use the stars and bars method. This method involves representing the balls as stars (*) and the dividers between boxes as bars (|). For 5 balls and 3 boxes, we need 2 dividers. The total number of symbols is 5 stars + 2 bars = 7 symbols. The number of ways to arrange these symbols is the number of ways to choose 2 positions out of 7 for the bars, which is calculated using combinations:[binom{5 + 3 - 1}{3 - 1} = binom{7}{2} = 21]Therefore, there are 21 ways to distribute the balls.

💡Okay, so Julian is working on a comic book, and I need to figure out how many frames there are in total. He has 143.0 frames per page, and there are 11.0 pages in his book. Hmm, that seems straightforward, but let me think through it carefully to make sure I get it right.First, I know that to find the total number of frames, I need to multiply the number of frames per page by the number of pages. That makes sense because if each page has a certain number of frames, then multiplying that by the total number of pages should give me the overall count.So, the number of frames per page is 143.0. That's a decimal number, but I guess it's just a way of representing the exact number of frames without any fractions. And the number of pages is 11.0. Again, that's a decimal, but it's essentially the same as 11 pages.Alright, so I need to multiply 143.0 by 11.0. Let me write that out:143.0 × 11.0Hmm, I can think of 11.0 as 11, since the decimal doesn't add any extra value here. So, it's just 143 × 11.Now, how do I multiply 143 by 11? I remember there's a trick for multiplying by 11. You can add the digits of the number together and place the sum in between them. But wait, that only works for two-digit numbers, right? Let me check.If I have a two-digit number like 23, multiplying by 11 would give me 253 because 2 + 3 = 5, and I place that in the middle. But 143 is a three-digit number. Does the same rule apply? Maybe I can try it.So, 143 × 11. Let's see:First, take the first digit: 1.Then, add the first and second digits: 1 + 4 = 5.Next, add the second and third digits: 4 + 3 = 7.Finally, take the last digit: 3.Putting it all together: 1 5 7 3, which is 1573.Wait, does that make sense? Let me verify by doing the multiplication the standard way to ensure I didn't make a mistake.So, 143 × 11:First, multiply 143 by 1 (the units place of 11):143 × 1 = 143Then, multiply 143 by 10 (the tens place of 11):143 × 10 = 1430Now, add those two results together:143 + 1430 = 1573Okay, that matches what I got using the trick. So, it seems like both methods give me the same answer, which is reassuring.But just to be absolutely sure, let me try another way. Maybe breaking down the multiplication:143 × 11 = 143 × (10 + 1) = (143 × 10) + (143 × 1) = 1430 + 143 = 1573Yep, that still gives me 1573. So, all three methods—using the trick, standard multiplication, and breaking it down—lead me to the same total number of frames.I think I'm confident now that the total number of frames in Julian's comic book is 1573.0. It's always good to double-check my work to make sure I didn't make any calculation errors, especially when dealing with decimals and multiplication.In summary, multiplying the number of frames per page (143.0) by the number of pages (11.0) gives me the total number of frames in the comic book, which is 1573.0.

💡Okay, so I have this math problem here that I need to solve. It looks a bit complicated at first glance, but I think if I break it down step by step, it should be manageable. Let me write it out again to make sure I have it right:Calculate the value of (1/25)b⁰ + (1/(25b))⁰ - (81^(-1/4)) - (-27)^(-1/3).And the options are:(A) 26/25(B) 1 3/25(C) 1(D) 24/25(E) 1/25Alright, so I need to simplify each term one by one and then combine them. Let me start with the first term.**First Term: (1/25)b⁰**Hmm, I remember that any number raised to the power of zero is 1, right? So b⁰ is 1, as long as b is not zero. So this term simplifies to (1/25)*1, which is just 1/25. Okay, that was straightforward.**Second Term: (1/(25b))⁰**Again, anything raised to the power of zero is 1, as long as the base is not zero. So regardless of what 1/(25b) is, as long as it's not zero, this term is 1. So this simplifies to 1.**Third Term: 81^(-1/4)**This one is a bit trickier. Negative exponents mean reciprocals, right? So 81^(-1/4) is the same as 1/(81^(1/4)). Now, 81 is a perfect fourth power because 3^4 is 81. So 81^(1/4) is 3. Therefore, 81^(-1/4) is 1/3. Got that.**Fourth Term: (-27)^(-1/3)**Okay, this one is a bit more complicated because of the negative base. Let me recall how exponents work with negative numbers. When dealing with fractional exponents, especially when the denominator is even, we can run into issues with real numbers, but since the denominator here is 3, which is odd, it should be okay.So, (-27)^(-1/3) is the same as 1/((-27)^(1/3)). Now, the cube root of -27 is -3 because (-3)^3 is -27. So, (-27)^(1/3) is -3, and thus 1/((-27)^(1/3)) is 1/(-3), which is -1/3.Wait, hold on. Is that correct? Let me double-check. If I take (-27) and raise it to the power of -1/3, it's the same as 1 over (-27)^(1/3). The cube root of -27 is indeed -3, so 1/(-3) is -1/3. Yeah, that seems right.**Putting It All Together**Now, let's substitute all the simplified terms back into the original expression:(1/25) + 1 - (1/3) - (-1/3)Simplify the signs:(1/25) + 1 - 1/3 + 1/3Wait a minute, the -1/3 and +1/3 cancel each other out. So those two terms add up to zero. That simplifies things.So now we have:(1/25) + 1Which is the same as 1 + 1/25. To add these together, I can express 1 as 25/25:25/25 + 1/25 = 26/25So, the value of the expression is 26/25.Looking back at the options, that's option (A).**Wait, Did I Make a Mistake?**Let me just go through it again to make sure I didn't make any errors.1. (1/25)b⁰ = 1/25 * 1 = 1/25. Correct.2. (1/(25b))⁰ = 1. Correct.3. 81^(-1/4) = 1/81^(1/4) = 1/3. Correct.4. (-27)^(-1/3) = 1/((-27)^(1/3)) = 1/(-3) = -1/3. Correct.Then, substituting:1/25 + 1 - 1/3 - (-1/3) = 1/25 + 1 - 1/3 + 1/3Yes, the -1/3 and +1/3 cancel, leaving 1/25 + 1 = 26/25.Hmm, seems solid. So, I think I'm confident that (A) is the correct answer.**Another Perspective**Just to be thorough, let me think about each term again.- The first term is straightforward: 1/25.- The second term is 1, which is a whole number.- The third term involves a negative exponent and a fractional root. I converted it correctly to 1/3.- The fourth term is a negative base with a negative fractional exponent. I had to recall how that works, but I think I handled it correctly by taking the cube root first and then inverting.Adding them up, the two fractional terms with 1/3 cancel each other, so they don't affect the sum. Then, adding 1/25 and 1 gives 26/25, which is 1.04 in decimal form. Looking at the options, 26/25 is option (A), which matches.I don't see any mistakes in my reasoning, so I think I'm good.**Final Check**Let me just write out the entire process in a more concise way to ensure clarity:1. (1/25)b⁰ = 1/252. (1/(25b))⁰ = 13. 81^(-1/4) = 1/34. (-27)^(-1/3) = -1/3So, substituting:1/25 + 1 - 1/3 - (-1/3) = 1/25 + 1 - 1/3 + 1/3Simplify:1/25 + 1 = 26/25Yep, that's correct. So, I'm confident that the answer is (A) 26/25.**Conclusion**After carefully simplifying each term and combining them, the value of the given expression is 26/25. This corresponds to option (A). I don't see any errors in my calculations, so I feel confident about this answer.**Final Answer**boxed{A}