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question:Let [z = frac{(-11 + 13i)^3 cdot (24 - 7i)^4}{3 + 4i},]and let w = frac{overline{z}}{z}. Compute |w|.
answer:First, we can simplify the expression for z as follows: begin{align*} z &= frac{(-11 + 13i)^3 cdot (24 - 7i)^4}{3 + 4i} &= frac{(-1331 + 4293i) cdot (20736 - 21504i)}{3 + 4i} &= frac{(-1331 cdot 20736 + 4293i cdot 20736) + (-1331 cdot -21504i + 4293i cdot 21504i)}{3 + 4i} &= frac{(-27669816 + 88975488i) + (28621984i + 92206512)}{3 + 4i} &= frac{-27669816 + 92206512 + (88975488i + 28621984i)}{3 + 4i} &= frac{64536704 + 117297472i}{3 + 4i} &= frac{(64536704 + 117297472i)(3 - 4i)}{(3 + 4i)(3 - 4i)} &= frac{(64536704 cdot 3 + 64536704 cdot -4i) + (117297472i cdot 3 - 117297472i cdot 4i)}{(3)^2 - (4i)^2} &= frac{(193610112 - 258147816i) + (351892416i - 469163888)}{9 + 16} &= frac{(-27553776 + 937447600i)}{25} &= -1102151.04 + 37497904i. end{align*}We can then compute overline{z} as the complex conjugate of z, which is -1102151.04 - 37497904i. Therefore, [w = frac{overline{z}}{z} = frac{-1102151.04 - 37497904i}{-1102151.04 + 37497904i} = frac{(-1102151.04 - 37497904i)(-1102151.04 - 37497904i)}{(-1102151.04 + 37497904i)(-1102151.04 - 37497904i)} = 1.]Then |w| = boxed{1}. The answer is: 1
question:A carton of ice cream contains 10 scoops each. Mary has 3 cartons of ice cream, one chocolate, one strawberry and one vanilla. Ethan wants one scoop of vanilla and one of chocolate. Lucas, Danny and Connor all want 2 scoops of chocolate. Olivia would like a scoop of vanilla and x of strawberry. Shannon wants twice as much as Olivia. How many scoops of ice cream will be left? If we know the answer to the above question is 16, what is the value of unknown variable x?
answer:Each carton of ice cream contains 10 scoops, so 3 cartons contain a total of 3 * 10 = 30 scoops. Ethan wants one scoop of vanilla and one scoop of chocolate, which means he will take 1 + 1 = 2 scoops. Lucas, Danny, and Connor all want 2 scoops of chocolate each, so they will take a total of 3 * 2 = 6 scoops. Olivia wants a scoop of vanilla and x scoops of strawberry, so she will take 1 + x scoops. Shannon wants twice as much as Olivia, so she will take 2 * (1 + x) = 2 + 2x scoops. The total number of scoops taken is 2 + 6 + (1 + x) + (2 + 2x) = 11 + 3x. To find the number of scoops left, we subtract the number of scoops taken from the total number of scoops: 30 - (11 + 3x) = 30 - 11 - 3x = 19 - 3x. We are given that there will be 16 scoops left, so we can write: 19 - 3x = 16. Solving for x, we get: 3x = 3. Dividing both sides by 3, we get: x = 1. The value of x is 1. 1 The answer is: 1
question:After slicing a loaf of bread into 12 pieces, Neeley's family consumed one-third of the slices for breakfast, and Neeley used 2 slices to make a sandwich for lunch. How many slices of bread are left?
answer:If Neeley's family consumed one-third of the slices for breakfast, then they ate 12/3 = 4 slices. After breakfast, there are 12 - 4 = 8 slices of bread left. If Neeley used 2 slices to make a sandwich for lunch, then there are 8 - 2 = 6 slices of bread left. 6 The answer is: 6
question:Karen had the giant box of crayons, containing twice as many crayons as were in Beatrice's box. But Beatrice's box contained twice as many crayons as were in Gilbert's box of crayons. And Gilbert had x times as many crayons as were in Judah's box of crayons. If Karen's box contained 128 crayons, There were 8 crayons in Judah's box. What is the value of unknown variable x?
answer:To solve this problem, we need to determine the value of x, which represents the number of times Gilbert's box of crayons contains compared to Judah's box. Let's break down the information given: Number of crayons in Judah's box: 8 Number of crayons in Gilbert's box: x * 8 Number of crayons in Beatrice's box: 2 * (x * 8) = 16x Number of crayons in Karen's box: 2 * (16x) = 32x Number of crayons in Karen's box: 128 We can set up the equation as follows: 32x = 128 To isolate x, we divide both sides of the equation by 32: 32x / 32 = 128 / 32 x = 4 The value of x is 4. 4 The answer is: 4