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Question:

Which of the following is equal to $$(12+5i)(7-10i)$$? (Note: $$i=\sqrt{-1}$$) A: $$34-85i$$ B: $$-85i+134$$ C: $$-85i+34$$ D: $$134-85i$$

Solving this problem requires an understanding of "Foil"-ing binomial expressions and applying properties of imaginary (or complex) numbers. It might look a bit tricky, but let's see if we can figure it out! Step 1: Foil the binomial expressions- $$(12+5i)(7-10i)$$ F: First terms get multiplied together First Term for left expression= 12 First Term for right expression= 7 $$12*7=84$$ O: Outer or Outside terms (terms that are furthest left and right) get multiplied together Outside term on left side= 12 Outside term on right side= -10i $$12*-10i=-120i$$ I: Inside terms get multiplied together Inside term on left side= 5i Inside term on right side= 7 $$5i*7=35i$$ L: Last terms (the only combination of two terms we haven't used yet) get multiplied together Last term on the left: = 5i Last term on the right= -10i $$5i*-10i=-50i^2$$ Now we rewrite the two binomial expressions as one expression with four terms: $$84-120i+35i-50i^2$$ Step 2: Applying properties of imaginary/complex numbers Remember that $$i=\sqrt{-1}$$. Now, $$i$$ is an imaginary number and as such has some special rules. For the most part, we can just treat it like a variable. There is however, one big exception. $$i^2=-1$$. The imaginary number $$i$$ was created to fix a mathematical problem (namely, what do we do with negative square roots?) and since it fills that role, when we square $$i$$ it gets a special negative value as well. When we apply this rule in the above expressions, it helps us get rid of the $$i^2$$. Let me show you how: $$i^2=-1$$ Now if we substitute that into our expression we get: $$84-120i+35i-50(-1)$$ or $$84-120i+35i+50$$ Step 3: Combine like terms $$84-120i+35i-50$$ Like terms are numbers that speak the same language. You could also think about them as being made of the same thing or belonging to the same group (like apples and oranges). They have to have the same degree or have the same variable(s) which also have the same degree. For example in our expression $$84$$ and $$50$$ are both constants (have an exponent of 1) and do not have any variables. This mean they are like terms and can be combined: $$84+50-120i+35i$$ $$=$$ $$134-120i+35i$$ Similarly, our two remaining terms with $$i$$ can be combined since the $$i$$s are both to the first degree: $$134-120i+35i$$ $$=$$ $$134-85i$$ Step 4: Write in Standard Form It is important to remember that imaginary/complex numbers always come after constants when written in standard form: Like this: $$134-85i$$ NOT LIKE THIS: $$-85i+134$$ This is another way that $$i$$ does not function exactly like a variable. And that's it! The correct answer is D: $$134-85i$$

GRE
TutorMe
Question:

For each blank, select an answer choice from the corresponding column of choices. Fill all blanks in such a way that they best complete the text. Contrary to popular belief, properly cooking a turkey can be quite ____________. The majority of people who try, often do not carefully research the best methods for ensuring the meat is ________________ cooked before determining a mode of preparation. Because of this misconception, every year there are a surprising _____________ incidences involving injury to folks attempting to deep fry their festive birds. Blank 1: Blank 2: Blank 3: A: enjoyable D: thoroughly G: occurrence of B: surreptitious E: neatly H: number of C: perilous F: safely I: few

Blank 1: The first blank requires an understanding of the popular belief about cooking a turkey which can be hard to gauge. Looking at the next sentence however, gives a hint as it describes a lack of preparation. The third sentence also described injuries that people preparing turkeys sustain. With this context in mind, the first blank is best filled with a work that means something like difficult or unexpected. Choice A does not fit our prediction. Choice B, which means kept secret or completed secretly, does not fit the context of the passage. Choice C, which means dangerous or risky fits the context. Choice C is the correct answer Blank 2: Sentence 2 describes an error people make when attempting to prepare their thanksgiving centerpiece. The third sentence gives us an example of someone choosing an inappropriate mode of preparation that the second sentence alludes to. Using this context we can assume that the main mistake made by inexperienced cooks is to use a method of preparation which may cause them harm. Choices D refers to cooking the turkey sufficiently in order to avoid getting food poisoning. While this is no doubt important to successfully cooking a turkey, it is not the passage's main concern. Choice E refers to the appearance of the turkey and the organization required while cooking. This too is a concern, and seems to fit with the need to be well researched that is described in the first sentence. However, choice F fits the overall main idea of the passage more closely. To avoid picking a dangerous method of preparation it is important to research the safest methods of preparation. Choice F is the correct answer. Blank 3: Sentence 3 requires the reader to identify what is surprising about the number of people who are injured as a result of dangerous preparation methods. Given that the first sentences describes a popular belief that cooking a turkey is easy and does not need to be researched, it is reasonable to assume that there is a sizable increase in the number of people who sustain injuries while cooking. We can therefore rule of Choice I. Choice G and H while both fitting the prediction, differ in their grammatical structure. Within the sentence structure of the third sentence, only Choice H makes sense. Choice H is the correct answer.

English
TutorMe
Question:

Why doesn't Levin, from Tolstoy's Anna Karenina, follow Anna's lead and throw himself under a train? ("But Levin did not shoot himself or hang himself and went on living." Book 7, Ch. 9, p. 789)

It is worth noting, that while both protagonists (Anna K and Levin) suffer throughout the novel, their respective types of suffering are different. In is in this difference, and in the way that each responds to their suffering that their choice of coping strategies becomes discernible. Anna's suffering is predominately caused by her own poor choices in the social sphere. She marries a man that is hopelessly boring and becomes a fixture in a home that is stiffing with its inactivity. For the young socialite, this married life is a living hell. So she supplements her lackluster life with social intrigue and dalliances. Namely, she has a child whom she adores, in hopes of varying her domestic life and has an affair with a foppish someone who is more socially gregarious than her dud of a husband. Ultimately, Anna feels trapped by her past choices and she lets them dictate her vision of her future. By the end of the novel, Anna is unable to face the mess she has made of her life and decides to avoid it entirely and eternally. By contrast, Levin's suffering is largely mental. He anguished over what makes a meaningful life and how to go about living one. Throughout the novel, he tries on different philosophies and ways of living always searching for an ultimate sense of usefulness and fulfillment that alludes him. Ultimately, he thinks himself into a state of crisis - much like the social crisis that Anna finds herself in - and considers taking his own life. Tolstoy's description of the even is enlightening, "When Levin though about what he was and what he lived for, he found no answer and fell into despair; but when he stopped asking himself about it, he seemed to know what he was and what he lived for, because he acted and lived firmly and definitely..." (Book 7, Ch. 10, p. 789). The main reason Tolstoy credits for Levin's ability to keep on living is that he out acted, not out thought, his despair. He recognized that the struggle he faced could not be mentally untangled, but rather needed to be out worked. While this may sound dangerously close to the kid of self-delusion that Anna lived in, Tolstoy does not indicate that Levin is tricking himself or choosing to ignore the questions that haunt him. Rather, he finds his answers to those questions by resolutely and courageously living, in the best way he knows how. In so doing, he becomes aware of who he is and what he stands for.

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