What Is the Distributive Property

What Is the Distributive Property SAT / ACT Prep Online Guides and Tips What is the distributive property? Did you go over the distributive property definition in school but still aren’t sure what it is or why it’s important? The distributive property is a key mathematical property you’ll need to know to solve many algebra problems. In this guide, we explain exactly what the distributive property is, why it’s important, when you should use it, what other math rules you need to know for it, and we also work through several examples so you can see the distributive property in action. What Is the Distributive Property? The distributive property, sometimes known as the distributive property of multiplication, tells us how to solve certain algebraic expressions that include both multiplication and addition. The literal definition of the distributive property is that multiplying a number by a sum is the same as doing each multiplication separately. In equation form, the distributive property looks like this: $a(b+c) = ab + ac$ (Remember, in math, when two numbers/factors are right next to each other, that means to multiply them.) Like many math definitions, the distributive property is easier to understand when you look at a few examples. Here’s a simple one: $$5 (2 +7)$$ Normally, if you had a problem like this, you’d add 2 and 7 together to get 9, then you’d multiply 5times 9 to get 45. This is the simplest way to solve the equation, and it also follows the order of operations, which tells you to simplify whatever is in the parentheses first before moving onto other operations like multiplication. Solving that equation using the distributive property would look like this: $$5 (2+7)$$ The distributive property means doing multiplication before the addition within the parentheses, so we’d distribute the 5 to both values within the parentheses: $$5(2) + 5(7)$$ Work out the multiplication: $$10 + 35$$ Then add the two numbers together: $$10+35=45$$ We get the same answer as we did solving the problem with the first method, which shows that the distributive property works. Now, why would you want to use the distributive property when it took longer than the first method? The distributive property comes in handy when you have terms within the parentheses that can’t be added together, such as this equation: ${3/4}(a + 2b)$. Because there are variables involved, there’s no easy way to simplify $a + 2b$. In these more complicated equations, the distributive property can help us get the equation into a form that makes it easier to simplify or solve. We’ll see examples of how to do this later on in this guide. 3 Key Rules Related to the Distributive Property When you’re using the distributive property, you’ll often have to use or be aware of other mathematical rules and properties in order to solve or simplify the equations. Here are three of the most important ones to know. Commutative Laws The commutative laws state that you can swap numbers when adding or multiplying and still get the same answer. So $x + y = y + x$ and $x(y) = y(x)$ These are likely intuitive for you by now, but they’re an important part of the distributive property, which wouldn’t work without them. You can use them when you need help simplifying certain equations in order to get them into a more workable form. Order of Operations When you have a complicated equation that looks like it can be simplified in multiple ways, the order of operations gives you the correct way to work through those operations. The acronym PEMDAS makes it easy to remember which operations to work on first. From first to last, here’s the order you should work out operations: Parentheses Exponents Multiplication and Division (do these at the same time, working left to right) Addition and Subtraction (do these at the same time, working left to right) The order of operations is important to know because you’ll often have to remember it when simplifying equations that include a lot of different operations. It can also help you determine whether to use the distributive property or not. Order of operations states the first step you should take when simplifying an equation is to work out whatever is in a parentheses set, but if what’s in the parentheses can’t be simplified, that’s a sign to use the distributive property. Quadratic Formula The quadratic formula states that, for $ax^2+ bx + c = 0$, the values of $x$ which are the solutions to the equation are given by: $$x={-b ±Ã¢Ë†Å¡{b^2-4ac}}/{2a}$$ When using the distributive property, you may be able to simplify some equations into the $ax^2 + bx + c = 0$ form so that you can use the quadratic equation to solve for $\bi x$. Distributive Property of Multiplication Example Problems In this section we go over three examples of simplifying problems using the distributive property. You’ll notice each of them contain variables in the parentheses, which is a key sign that the distributive property is needed. Example 1 $$\bo4\bi x(\bo5\bi x + \bo6) = -\bo7$$ First, we’re going to distribute $4x$ to both $5x$ and 6. $$4x(5x) + 4x(6) = -7$$ Now, multiply those out: $$20x^2+ 24x = 7$$ Add 7 to both sides: $$20x^2+ 24x +7 = 0$$ This equation is now in the proper formula to solve for $x$ using the quadratic formula (x would equal $-0.7$ and $-0.5$), or you might be able to keep the equation in that form if you were just being asked to simplify it. Example 2 $$\bo3\bi x(\bi x-\bo4) + \bo5(\bo4\bi x + \bo6)$$ For this equation, there are two sets of parentheses, so we need to use the distributive property twice. Distribute the 3x to its set of parentheses and the 5x to its set of parentheses: $$3x(x) + 3x(-4) + 5(4x) + 5(6)$$ Multiply it out: $$3x^2- 12x + 20x^2+ 30$$ Add the two $x^2$ terms together to simplify $$23x^2- 12x + 30$$ Example 3 $$-\bo7(\bi x + \bo4) + \bo8(\bo2 - \bo4\bi x)$$ This example is a bit trickier because the 7 has a negative sign in front of it. When the value just outside the parentheses is negative, the negative sign must be distributed to each term within the parentheses. Distribute the -7 to its set of parentheses and the 8 to its set of parentheses: $$(-7)(x) + (-7)(4) + (8)(2) + (8)(-4x)$$ Multiply those out: $$-7x -28 + 16 - 32x$$ Now simplify: $$-39x - 12$$ Summary: What Is the Distributive Property Definition? What is distributive property? The distributive property of multiplication states that $a(b+c) = ab + ac$. It’s often used for equations when the terms within the parentheses can’t be simplified because they contain one or more variables.Using the distributive property, you can simplify or solve equations that would otherwise be difficult to work with. When using the distributive property, remember to distribute negative signs if they’re in front of the parentheses, and keep in mind other important math rules, such as the quadratic formula, order of operations, and commutative properties. What's Next? Are you learning about logarithms and natural logs in math class? We have a guide on all the natural log rules you need to know. What is dynamic equilibrium and what does it have to do with rusty cars? Find out by reading ourcomplete guide to dynamic equilibrium. Rational numbers are another important math concept you should understand.Read our guide on rational numbers for everything you need to know about them!

The Roman Tetrarchy and the Rule of Four

The Roman Tetrarchy and the Rule of Four The word Tetrarchy means rule of four. It derives from the Greek words for four (tetra-) and rule (arch-). In practice, the word refers to the division of an organization or government into four parts, with a different person ruling each part. There have been several Tetrarchies over the centuries, but the phrase is usually used to refer to the division of the Roman Empire into a western and eastern empire, with subordinate divisions within the western and eastern empires. The Roman Tetrarchy Tetrarchy refers to the establishment by the Roman Emperor Diocletian of a 4-part division of the empire. Diocletian understood that the huge Roman Empire could be (and often was) taken over by any general who chose to assassinate the emperor. This, of course, caused significant political upheaval; it was virtually impossible to unite the empire. The reforms of Diocletian came after a period when many emperors had been assassinated. This earlier period is referred to as chaotic and the reforms were meant to remedy the political difficulties that the Roman Empire faced. Diocletians solution to the problem was to create multiple leaders, or Tetrarchs, located in multiple locations. Each would have significant power. Thus, the death of one of the Tetrarchs would not mean a change in governance. This new approach, in theory, would lower the risk of assassination and, at the same time, made it nearly impossible to overthrow the entire Empire at a single blow. When he split up the leadership of the Roman Empire in 286, Diocletian continued to rule in the East. He made Maximian his equal and co-emperor in the west. They were each called Augustus which signified that they were emperors. In 293, the two emperors decide to name additional leaders who could take over for them in the case of their deaths. Subordinate to the emperors were the two Caesars: Galerius, in the east, and Constantius in the west. An Augustus was always emperor; sometimes the Caesars were also referred to as emperors. This method of creating emperors and their successors bypassed the need for approval of emperors by the Senate and blocked the power of the military to elevate their popular generals to the purple. [Source: The City of Rome in late imperial ideology: The Tetrarchs, Maxentius, and Constantine, by Olivier Hekster, from Mediterraneo Antico 1999.] The Roman Tetrarchy functioned well during Diocletians life, and he and Maximian did indeed turn over leadership to the two subordinate Caesars, Galerius and Constantius. These two, in turn, named two new Caesars: Severus and Maximinus Daia.  The untimely death of Constantius, however, led to political warring. By 313, the Tetrarchy was no longer functional, and, in 324, Constantine became sole Emperor of Rome.   Other Tetrarchies While the Roman Tetrarchy is the most famous, other four-person ruling groups have existed through history. Among the best-known was The Herodian Tetrarchy, also called the Tetrarchy of Judea. This group, formed after the death of Herod the Great in 4 BCE, included Herods sons.

Autism Spectrum Disorder Research Paper Example | Topics and Well Written Essays - 1000 words

Autism Spectrum Disorder - Research Paper Example ASD varies in individuals concerning its onset, with cases that are from birth while others develop in later life. In the area of learning, impairment makes the affected child who could be, for example, be in the fifth grade, to have the reading level of a second grade level child. In the area of socialization, other children start to portray withdrawal symptoms, where they prefer to be alone, and seem uninterested in the games or activities of the people around them. This affects their communicational development and their learning ability (Tidmarsh and Volkmar, 2003) because socialization is a very important aspect of childhood development, since children learn some aspects of behavior through imitating what other children are saying or doing (Tidmarsh and Volkmar, 2003). These children with ASD lack this chance of developing if the situation is not addressed through encouraging them to interact with other children (Volkmar, 2012). Some children with ASD also show signs of uncontro llable fits of temper, great resistance to change and are oversensitive or under sensitive to sounds or sights, while others too are prone to epileptic seizures and fits, anxiety and depression. Others will portray the lack of interest in the people around them, avoiding eye contact and probably only showing interest in toys and other objects or in a certain subject. Some of the victims of ASD speak very few or no words at all, but use pictures and signs to communicate, while others are preoccupied with certain objects or items. Many victims of ASD portray a consistent adherence to some specific rules and rituals . It is also worthwhile to note that autism affects boys more than girls with the ratio being that out of every five people with the disorder, four are male (Volkmar, 2012). Research on the possible causes of autism has been done extensively and is continuing, but no clear causes have been established but scientists have been able to identify some various types of genes tha t are associated by the disorder and there also have been noticeable relationship with the disorder among close family members, indicating that genetics is a major factor (Tidmarsh and Volkmar, 2003). Research has shown that an identical twin has a 90% chance of getting ASD if the other twin has the disorder, further advancing the theory about genetics (National Institute).It has also been observed that due the presence of high levels of serotonin and other transmitters of information to the brain, ASD could be caused by their interfering with normal brain formation starting from fetal stage to later life. These substances have been found to occur in higher than normal levels in people with ASD. ASD has no known cure but therapeutic and behavioral interventions can be and have been used to bring a remedy to some of the specific symptoms observed and, where applied there have been great improvements. While symptoms may vary, it is very important that they be noted as early as possibl e so that ways to manage the disorder can be enhanced earliest possible (Tidmarsh and Volkmar, 2003). The following methods would be helpful to assist or educate a victim of ASD in the best way possible. I would emphasize on the screening and proper diagnosis of his

Self Assessment Paper Essay Example | Topics and Well Written Essays - 1500 words

Self Assessment Paper - Essay Example 1). Therefore, the implication of this score on one’s future employer where managers believe in theory Y is that they could design motivational strategies that would empower employees like me to work with proper recognition and acknowledgement. Personal Values The top three personal values were identified in these areas with the following scores: family (200), social (200) and intellectual (200). The lowest personal value is spiritual (70). These indicates how I regard these values are paramount in one’s life. The implication of this is crucial in the organization where I would be part of to balance time spent for work and family life, as well as in social and intellectual endeavors. Intolerance for Ambiguity My score for this assessment is 52. The mean that was noted in the interpretation was actually 49; therefore, the score that I generated was actually a little bit higher than the mean. This could indicate a higher tolerance for ambiguity or a slightly high preponde rance for risky endeavors or delving into the unknown. The implication of this is that future organizations could tap my willingness and ability to adapt and adjust to turbulent situations; as such, I am not easily swayed or detracted from facing challenges and adversities. Two-Factor Profile My MF score is 40; while my HF score is 30. This means that I regard greater importance on motivating or satisfying factors than in hygiene or dissatisfying factors according to Herzberg’s two-factor theory (Schermerhorn, Osborn, Uhl-Bien, & Hunt, 2012). This also means job content is more important to me than job context. The implication of this result is that managers should focus on motivators through job content if they want to influence or motivate employees like me. Your Personality Type The score that was generated for this assessment is 35. This is indicative of a Type A personality which was described to be hurried and competitive. The implication of this as one is a member of a n organization is that my personality type should complement that of others; otherwise, if all members of the organization are all competitive, there could be increased pressure to exhibit a stipulated level of performance that could be stressful and extremely challenging. Time Management Profile My score for this assessment is 4. As indicated, the higher the score, the closer one’s behavior matches the recommended time management guidelines. In this regard, one strongly believes that my score is an indication of the need to improve on time management to coincide with those prescribed by the organization or institution that one would eventually serve. Week 2: Are You Cosmopolitan? My score in this assessment is 23. This means that that I am within the mixed orientation (between being manifesting a cosmopolitan orientation and a local orientation) or one exhibiting and identifying a career profession and that of an employing organization. Thus, the implication is that being at mid-point, I could assume both preferences and orientation that the two extremes apparently exhibit.

The Recluse Essay -- Literary Analysis

Wordsworth suffers solitude, even as he celebrates it. Alone, the poet can explore his own consciousness; it exists at both poles of the notion of ‘emotion recollected in tranquillity’, and is the dominant developmental mode of Wordsworth’s childhood as depicted in The Prelude (1805). Independence is what is exalted in his introduction to that poem: he greets the ‘gentle breeze’ as a ‘captive†¦ set free’ from the ‘vast city’ which has been as a ‘prison’ to his spirit. The oppression of city living is alleviated in this opening reacquisition of isolation; the relief is evident: ‘I breathe again’, ‘that burthen of my own unnatural self [is shaken off], /The heavy weight of many a weary day/ Not mine, and such as were not made for me’. In this, the commencing statement of his autobiography, the independence of solitude is represented as the essential quality of his poetic felicity. T he ‘egoistical sublime’ observed by Keats is manifest in this poetry in a separation from other men, rather than in that of a Byron, whose narrators’ egotisms are evinced by their social interactions. Wordsworth’s company is nature; his sister, his wife, his children exist as assimilations rather than relationships. The sister of Lines Written a Few Miles Above Tintern Abbey, is conjured into independence in the final paragraph, so as to exist as a previous self: ‘For thou are with me’, he suddenly reveals, ‘and in thy voice I catch/ The language of my former heart’. She is externalised when poetically useful; and it is by this externalisation that Wordsworth is able to avert and diminish his poem’s undercurrent doubts. ‘This prayer I make/ Knowing that Nature never did betray/ The heart that loved her’, has a contrary traction as a plea intimating des... ...this as his essential condition, but it is worth observing that ‘recluse’ does not imply total isolation. Wordsworth’s solitude, as he left childhood, was never again to be absolute; for as consciousness developed, so did his capacity to apprehend himself, in language, so even alone he could not be alone without self-intercourse, mediated by language. His solitude was necessary for his vocation, but his vocation trespassed on that solitude; for to be a poet is to cast experience away from the self: even in egotism, isolation is disrupted by the projection of an audience. Works Cited Gil, Stephen ed. William Wordsworth: The Major Works (OUP 1984) Hartman, Geoffrey Wordsworth’s Poetry 1787-1814 (Yale University Press 1971) Morgan, Monique R. ‘Narrative Means to Lyric Ends in Wordsworth’s Prelude’ (Narrative, Volume 16, Number 3, October 2008, pp. 298-330)

Mrs. Aesop by Carol Ann Duffy by Andrew Banks Essay

1. The poem â€Å"† is part of Carol Ann Duffy’s collection of poems, titled â€Å"The World’s Wife†. In this collection, Duffy wishes to highlight the fact that women have long been ignored and silenced throughout history. This is why all the poems in the collection are written from a female perspective. Duffy has created a literal version of an old saying, â€Å"behind every great man there is an even greater woman†. One of the poems in the collection, ‘Mrs. Aesop’, tells the story of a wife who is tired of her sermonizing, tedious husband, known as Aesop. Aesop was a storyteller who lived around the sixth century BC, in Greece. Many historical details surrounding him are missing, but it is thought that he was first a slave on the island of Samos and his fables came to be in a collection known as â€Å"Aesopica.† â€Å"Mrs. Aesop† draws on the fables to describe Aesop’s wife’s discontent and unhappiness, the p oem emasculating her husband. The major theme of this poem is to make apparent Mrs. Aesop’s transformation from the classic recessive wife with a dominant husband, to an empowered and confident woman that was able to have the last word. This is shown by lines such as, â€Å"That shut him up. I laughed last, longest.† This appears in the poem after Mrs. Aesop has mocked her husband’s impotence, with lines like â€Å"I gave him a fable one night/ about a little cock that wouldn’t crow†¦Ã¢â‚¬  mocking his masculinity whilst clearly referring to his genitalia. 2. Allusions are some of the many literary devices Duffy includes in her works to better deliver the messages of the female protagonists in â€Å"The World’s Wife.† An allusion is a brief and indirect reference to a person, place, thing or idea of historical, cultural, literary or political significance. For example, one would be making a literary allusion when stating, â€Å"I do not approve of such a quixotic idea.† Quixotic takes on the meaning of foolish and impractical, derived from Cervantes’s â€Å"Don Quixote†, a story about the misadventures of a doltish night and his cohort Sancho Panza. Many allusions are ones we use in our daily speech, such as Achilles’ heel – A weakness a person may have. Achilles was invulnerable excepting his heel or Achilles tendon. Pygmalion – Someone who tries to fashion someone else into the person he desires, originating from a myth adapted into a play by George Bernard Shaw. Casanova – a man who is amorous to women, based on the Italian adventurer.McCarthyism – modern witch hunt, the practice of publicizing accusations without evidence, made after Joseph McCarthy. Some allusions in the literature include when the character Horatio from Shakespeare’s â€Å"Hamlet† said â€Å"A mote it is to trouble the mind’s eye. In the highest and palmy state of Rome, A little ere the mightiest Julius fell, The graves stood tenantless and the sheeted dead Did squeak and gibber in the Roman streets†¦Ã¢â‚¬  (I.i.111-115) Here, Horatio is making a reference to the historical figure of Julius Caesar, in addition to one of Shakespeare’s earlier plays titled â€Å"Julius Caesar.† Another time when allusions are used are in songs, such as when Nirvana made their classic, â€Å"Scentless Apprentice†: Like most babies smell like butter His smell smelled like no other. He was born scentless and senseless He was born a scentless apprentice. this allusion is to Patrick Sà ¼skind’s literary work Perfume. The scent Nirvana is alluding to is actually the blood of the protagonist’s twenty killing victims. 3. When Duffy uses allusions in Mrs. Aesop, she mainly uses them in the text to show Mrs. Aesop’s unhappiness with her husband In the first line, Mrs. Aesop says â€Å"By Christ, he could bore for Purgatory†. This is an allusion to Christianity, with purgatory being the place after death where souls go to be cleansed of their sins. The implication here is that Aesop could make this experience even worse. Later in the first stanza, Duffy alludes to one of Aesop’s fables, when Mrs. Aesop puts her own twist on the lineâ€Å"A bird in the hand is worth two in the bush† changing it to â€Å"the bird in his hand that on his sleeve.† By adding to his work in such a way, Mrs. Aesop is disrespecting both her husband and his work, revealing the emotion she had kept bottled up for some time. Lines such as â€Å"a tortoise, somebody’s pet,/ creeping, slow as marriage, up the road,† are a clear allusion to the tale of the Tortoise and the Hare. Mrs. Aesop uses the tortoise and hare to describe the agony of her marriage. With her cynical view Mrs. Aesop shows to her, the fable is nothing more than the reflection of a terrible marriage. When Mrs. Aesop says â€Å"I’ll cut off your tail, all right, I said, to save my face.† this is another reference to her own suppression by her husband and many other wives. This is so because the line alludes to an incident in America in 1993 when a frustrated wife sliced off her husband’s genitals in a moment of crazed revenge. Mrs. Aesop takes on a similar path to gain the upper hand on her husband, by disrespecting and revealing her true feelings about her husband. Allusions are a key literary device used to show Mrs. Aesop’s transition from a â€Å"traditional† oppressed wife to a dominant, independent woman.

Summary and Background on The Vagina Monologues

A night of theater can be much more than getting dressed up to watch a Rodgers and Hammerstein revival for the umpteenth time. Theater can be a voice for change and a call to action. Case in point: The Vagina Monologues. Playwright and performance artist Eve Ensler interviewed over 200 women from a wide range of ages and cultural backgrounds, many of whom bared their proverbial souls by responding to questions like, What would your vagina say if it could talk? and, If you could dress your vagina, what would it wear? Origins and V-Day In 1996, The Vagina Monologues began as a one-woman show, a series of character-driven pieces. Almost like poetry, each soliloquy reveals a different womans experience with topics like  sex, love, tenderness, embarrassment, cruelty, pain, and pleasure. As the show gained popularity, it was performed by an ensemble of actresses. Politically active theaters and college campuses began staging productions of the monologues, which helped launch a global movement known as  V-Day. What Is V-Day? V-Day is a catalyst that promotes creative events to increase awareness, raise money and revitalize the spirit of existing anti-violence organizations. V-Day generates broader attention for the fight to stop violence against women and girls. Anti-Male Sentiments? When college students are asked to raise their hands if they are feminists, often  only one or two students raise their hands. The women students who dont raise their hands misguidedly explain that they dont hate men, whereas many uninformed men believe that a necessary prerequisite for membership in feminism is womanhood. Sadly, while feminism is understood to mean equality for the sexes or the empowerment of women, it seems that many believe feminism is anti-male. With that in mind, it is easy to see why many assume that The Vagina Monologues is an angry rant of naughty words and feverish male-bashing. But Ensler is clearly raging against violence and oppression rather than men in general. V-Men, a digital section of V-Day where male writers and activists speak out against misogynist violence, is further proof that Enslers work is man-friendly. Powerful Moments The Flood: This monologue, based on a conversation with a 72-year old woman, combines humorously erotic dream imagery with the pragmatic, worldly views of a tough, outspoken old gal. Picture your elderly great Aunt talking about down there, and youll get an idea of this monologues potential. During her HBO special, Ensler has great fun with this character.My Village Was My Vagina: Powerful, sad, and all-too-relevant, this is absolutely the most haunting of the monologues. This piece is in honor of the thousands of victims from rape camps in Bosnia and Kosovo. The monologue alternates between peaceful, rural memories and images of torture and sexual abuse.I Was in the Room: Based on Enslers personal experience watching the birth of her grandchild, this is arguably the most touching and optimistic monologue. This scene captures the joy and mystery of labor, in all its glorious and graphic detail. The Controversial Monologue Sure, the whole show is controversial. Theres shock value simply in the title. Still, one particular monologue involves two accounts of molestation. The first incident occurs when the character is 10. In that account, she is raped by an adult male. Later in the monologue, she describes a sexual experience with an adult woman when  the speaker is only 16. This monologue upsets many viewers and critics because it presents a double standard. The first case of molestation is accurately nightmarish, whereas the second case is portrayed as a positive experience. In an earlier version, the lesbian encounter took place at the age of 13, but Ensler decided to adjust the age. Because she generated the monologues from real-life interviews, it makes sense to display what she learned from her subject. However, considering the mission statement of V-Day, its hard to fault directors or performers for omitting—or perhaps revising—this particular monologue. Other Ensler Plays Although The Vagina Monologues is her most famous work, Ensler has penned other powerful works for the stage. Necessary Targets: A gripping drama that depicts two American women journeying to Europe to help Bosnian women share their tragic stories with the world.The Treatment: Enslers most recent work delves into the moral questions of torture, power, and the politics of modern warfare.