Adding and subtracting rational expressions with like denominators is similar to adding and subtracting fractions with like denominators. It is a three-step process: first add the numerators, then simplify the numerator, then simplify the result by factoring. Since subtraction is the inverse of addition, subtraction is a matter of adding the inverse of the numerator.

Suppose the problem is 4x/5 + 6x/5. Since the numerators have a variable in common, 4x +6x can be added as 10x. Then (10x)/5 has the common factor of 5 in both the numerator and denominator. The simplified result is (2x)/1, or 2x. Suppose there were no variable in the expression. It would then be a fraction, so that 4/5 +6/5 would equal 10/5 or 2.

Suppose the problem is (8y^{2})/(y +3) + (2y^{2})/(y+3). The numerators have variables, and so do the denominators. In this case, the denominator in each expression is the same, so the numerators can be added as 8y^{2} +2y^{2} =10y^{2}. The expression is then (10y^{2}/(y +3). There are no factors in common, so it is already in simplest terms.

Suppose the problem is (x^{2}-4x-10)/(x-7) +(x-18)/(x-7). The denominator is the same for both rational expressions, so the numerators can be added. Combining like terms, (x^{2}-4x+x -10 -18), the x^{2} is the only exponent. Then, -4x +x equals -3x, for the variables. Then, (-10 -18) equals -28, so the numerator is now (x^{2}-3x-28), which can be factored as (x +4)(x-7). Using FOIL to check, x·x equals x^{2}, 4x-7x equals -3x, and (-7·4) equals -28. The numerator [(x +4)(x-7)] can now be placed over the common denominator (x-7). The (x-7) in the numerator and denominator cancel out, so the answer in simplest terms is (x +4).

As subtraction is adding the inverse, the process is very similar to addition of rational expressions with like denominators. The numerator is subtracted, then the rational expression is simplified. For example, the first step to solve[(4m +5)/(m-1)] – [(2m-1)/(m-1)] is to combine the numerators as [(4m +5) –(2m -1)] or 4m-2m +5 +1 or 2m +6. The expression 2m +6 is not in simplest terms, as it can be factored as 2(m +3). Then using the common denominator, the entire solution is [2(m +3)]/(m-1).

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]]>Shakespeare’s plays have been performed all over the world, translated into many other languages, and sets have been adapted beyond the traditional stage settings. In addition to movies made from the plays, there have been movies based on Shakespearean themes, operas and other musical works set to Shakespeare, as well as books, dance, and other forms of visual arts.

Shakespeare’s plays were staged elaborately when they were originally performed, with elaborate scenery, fencing, dance, music, sound effects, and even fireworks. The Globe Theatre itself burned in 1613 when a cannonball caught fire. *Hamlet* has been reset to Haiti, *Macbeth* to World War I, and *Richard III* to Nazi Germany. *The Taming of the Shrew* was reset as the Broadway musical *Kiss Me Kate*. *Romeo and Juliet *was reset to rival gangs in New York City in the Broadway musical *West Side Story*, later to be made into a movie.

There have been more than 400 movie and TV adaptations of Shakespeare plots. Many of them have been based on *Romeo and Juliet*, besides the film adaptations such as the 1968 film *Romeo and Juliet* directed by Zeffirelli and Baz Luhrmann’s 1996* Romeo + Juliet*. The movie *Shakespeare in* *Love* is a retelling of Shakespeare writing the play, and* Romeo Must Die* (2000) uses mixed martial arts to tell the story. There have also been many adaptations of other plays or Shakespearean concepts. The movie *10 Things I Hate About You* is based on The *Taming of the Shrew*. *Hamlet* has been adapted to *The Lion King*, *Let the Devil Wear Black*, and *Rosencrantz and Guildenstern Are Dead*. Science fiction based on Shakespeare includes the* Star Trek* episode “The Conscience of the King”, a *Doctor Who* episode, “The Shakespeare Code”, and the android love story *Romie-0* *and Julie-8.*

*A Thousand Acres* by Jane Smiley and *The Serpent’s Tooth* by Diana L. Paxson are among the many books based on* King Lear*. *The Daughter of* *Time* by Josephine Tey is connected to *Richard III*, *Brave New World* by Aldous Huxley to *The Tempest*, and *The Talented Mr. Ripley* by Patricia Highsmith to *Macbeth*. Every year brings new books and stories, as Shakespeare’s themes are universal.

Since music and dance were such a part of Shakespeare’s plays, many ballets have been staged to tell the stories, such as *Romeo and Juliet* with music by Prokofiev and by Tchaikovsky, *A Midsummer Night’s Dream* by Mendelssohn, and a one-act version, The Dream by Ashton, along with many others. Verdi scored the dramatic operas *Otello* and *Macbeth*, as well as the comedy *Falstaff*, based on *The Merry Wives of Windsor*. Vaughn Williams composed the opera *Sir John in Love*, also based on *The Merry Wives of Windso*r. Many paintings were inspired by scenes and characters from Shakespeare, including work by William Blake, Millais, and others.

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]]>William Shakespeare (1564-1616) is known as the greatest writer in the English language and dramatist to the world. He wrote over 35 plays, including histories, comedies, and tragedies. His sonnets are a collection of 154, published in 1609. They contain some of the most beautiful love poetry ever written.

Shakespeare wrote historical dramas that appeared in the First Folio collection. Most of these plays dealt with English history, such as *King John* and *Henry VIII*, and a complete series of plays about the Wars of the Roses: *Richard II*, *Richard III*, *Henry IV* parts I and II, *Henry V*, and *Henry* *VI*, parts I, II, and III. As Shakespeare was living during the reign of Elizabeth I, the daughter of Henry VIII and Anne Boleyn, he wrote to advance the Tudor cause and the Tudor dynasty. Richard III is seen as a monster, and the just end of the York line, *Henry VIII* ends with the birth of Elizabeth I. Many of his plays that he called comedies and some of his plays that he called tragedies are also historical in nature.

In Shakespeare’s time, comedies were plays with happy endings, sometimes with happy marriages, and they were more lighthearted in tone than the histories or tragedies. Some of those comedies include *A Midsummer Night’s Dream*, *As You Like It*, *The Merry Wives of Windsor*, *The Two* *Gentlemen of Verona*, *Much Ado about Nothing,* and *Twelfth Night.* Some of his later plays, such as *The Tempest*,* Cymbeline*, and *A Winter’s Tale* came to be known as romances. Plays such as *All’s Well that Ends Well* and *Measure for Measure* were also known as problem plays, as they mixed humor and tragedy in an unusual way.

Tragedies present characters in plots that blend death, madness, and murder. Some of Shakespeare’s great tragedies were based on historical Roman figures, such as *Antony and Cleopatra*, *Julius Caesar*, and *Coriolanus*. *Romeo and Juliet* and *Othello* used Italian narratives as sources, and *King Lear* was derived from tales about ancient kings. Hamlet was a prince in Denmark, and *Macbeth* was believed by many to carry a curse, as murder, intrigue, and deception are throughout the play.

The Sonnets are a collection of 154 poems, published in 1609. The first 126 sonnets were addressed to a young man (The Fair Youth), and the last 28 were addressed to a woman (The Dark Lady). From the time that they were first printed, readers wondered to whom they were dedicated. Were the Fair Youth, the Dark Lady, and the Rival Poet real people, characterizations, or aspects of Shakespeare’s culture? Critics have specified many different opinions on their nature.

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]]>During solar eclipses, more is revealed about the features in the sun’s corona. The corona is not normally visible to the naked eye, so scientists began studying it long before satellites were launched into space. In addition, observations during a solar eclipse brought important early proof of Einstein’s theory of relativity.

People have viewed solar eclipses throughout history. A megalithic monument in Ireland shows spiral carvings on rocks, called petroglyphs, which code the positions of the sun at various times. One of those may correspond to a total solar eclipse in 3340 BCE. In Ancient China, solar eclipses were thought to be predictions of doom to the Emperor who ruled at the time, and Chinese astrologers kept careful records of them. They thought that the sun was devoured by a huge green dragon that lived somewhere in the heavens. To scare away the dragon, the Chinese people beat pots and drums to make noise and scare away the dragon. In 2134 BCE, they described an eclipse in a phrase that translated as “the sun and the moon did not meet harmoniously.”

In Babylon, astronomers kept very careful records of the sun and moon’s position, so that they eventually used a mathematical formula to predict eclipses. Some of the eclipses they observed included one in 1063 BCE that “turned day into night”, and a famous one in 763 BCE, recorded by Assyrians in the ancient city of Nineveh. The ancient Greeks recorded eclipses as well as developing geometric predictions of when and where both lunar and solar eclipses would occur. By 200 CE, eclipses were a predictable scientific fact rather than an unpredictable sign of cosmic doom for much of the world.

**Early Observations of the Solar Corona**

The earliest observations of solar prominences were in 334 CE, and the corona was described 600 years later. By the 1700s, the corona was named as a crown around the sun, as part of the sun that could only be seen during the eclipse. The first wet plate photograph of a solar eclipse was taken in 1860, and many other discoveries followed.

A new line was found in the sun’s spectrum during an eclipse in 1868, and the chemist named it “helium” after the ancient Greek word for the sun, Helios. The element helium was not formally discovered on earth until 30 years later. On May 29, 1919, Sir Arthur Eddington carried out an experiment during the solar eclipse that was an early proof of Einstein’s theory of relativity. The positions of stars during the solar eclipse were carefully measured and compared with measurements of those same stars in their normal positions (not during a solar eclipse). Those careful measurements showed that space-time was warped by the gravity of the sun, exactly as predicted by Einstein’s equations. Eddington’s findings have been replicated in experiments elsewhere in the world, and scientists will also use the solar eclipse of August 21, 2017 as another opportunity to simulate his famous experiment, 98 years later.

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]]>On Monday, August 21, 2017, many people in North America will be able to see either a partial solar eclipse or a total solar eclipse, depending on where they live. Parts of South America, Africa, and Europe will also see a partial solar eclipse.

During a solar eclipse, the moon passes between the earth and the sun. It blocks either all of the light from the sun or part of the light from the sun during the time it takes for the moon to travel across the face of the sun. The darkest portion of the moon’s shadow, the umbra, falls along a narrow path on the earth, for a total solar eclipse. The penumbra, or half of the moon’s shadow, falls on either side of the umbra. The sun is much larger than the earth or the moon, so it produces both umbra and penumbra.

The path of totality is a 70 mile wide region that stretches from Lincoln Beach, OR through Oregon, Idaho, Wyoming, Montana, Nebraska, Iowa, Kansas, Missouri, Illinois, Kentucky, Tennessee, Georgia, and North and South Carolina. Other regions north and south of the path of totality will see a partial eclipse. Although the sun appears to be covered for about two minutes, the entire eclipse from start to finish takes about two hours. Many different websites on the Internet show whether the eclipse will be total or partial based on the location of the city.

A partial eclipse begins when the sun appears to be partially blocked by the moon. That phase can last over an hour, and the sun looks like a partially-obscured crescent. The next phase before totality is often called the “diamond ring” because a single brilliant point of light shows at one place, while the corona of the sun forms a ring. The phase just before totality is referred to as “Baily’s beads” which are shining points of sunlight that are formed from the hills and valleys of the moon as it almost obscures the sun. During totality, the moon covers the entire disk of the sun and only the corona shows. In the final stages, the growing crescent sun shows as the moon moves away from blocking it. If the eclipse is a partial eclipse, none of the phases just before totality show.

Do not look at the sun directly without proper eye protection. Regular sunglasses do not provide enough protection, so special “eclipse glasses” are readily available. There are also many ways to view the eclipse indirectly, such as by using a sun funnel to reflect light coming from a telescope, projecting the image of the eclipse onto a screen, or use a pinhole projection.

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]]>The process of multiplying and dividing rational expressions is similar to multiplying and dividing rational numbers. In order to multiply, first multiply the numerators, then the denominators, and then simplify the expression. Division is the inverse of multiplication, so multiply by the reciprocal (or inverse) of the divisor, and then simplify the expression.

Rational numbers are also known as fractions. The fraction ¾ is a rational number, as is the fraction 3/5. The numerator 3 is in a ratio to the denominator 4. To multiply ¾ by 3/5, multiply the numerators 3·3, and then multiply the denominators 4·5, to result in the new fraction or ratio of 9/20. The fraction 9/20 is in the simplest form, because there are no factors common to both 9 and 20 except for 1. Suppose one ratio were a/9 and another were 7/10. The process of multiplication would be similar, so that a/9 times 7/10 would equal 7a/90.

Suppose the problem were (5a^{3})/4 times 2/ (5a). The numerator is 5a^{3}·2 or 10a^{3}, and the denominator is 4 ·5a or 20a. The new expression, (10a^{3})/20a can be simplified to ½ ·a^{2} or a^{2}/2. If the problem were 4/ (5x^{2}) ·(x-2)/ (2x^{3}), the new numerator would be 4(x-2) and the new denominator would be (5x^{2}) · (2x^{3}) or 10x^{5}. The new expression is then [4(x-2)]/ (10x^{5}), which is not in simplest form. The fraction 4/10 can be simplified to 2/5, so the expression in simplest form is [2(x-2)]/ (5x^{5})].

Remember that dividing rational numbers is the same as multiplying by the reciprocal of the divisor, so that 4/5 ÷2/3 is the same as 4/5 ·3/2, so that 4·3 equals 12 and 5·2 equals 10. Since 2 is a common factor in both the numerator and denominator, the fraction in simplest terms is 6/5. There are no common factors to both 6 and 5 except for 1 so the fraction is in simplest terms.

Similar to dividing rational numbers, when dividing rational expressions, also multiply by the reciprocal of the divisor. Therefore (8n^{5})/3 ÷ (2n^{2})/9 becomes (8n^{5})/3 ·9/ (2n^{2}). Multiply [(8n^{5}) · 9], the numerator and [3· (2n^{2})]. However, the resulting expression (72n^{5})/ (6n^{2}) is not in simplest terms. First, factor out the common coefficients, so that 12 is left. (The number 72 divided by 6 equals 12.) Next factor out the common variables, so that n^{5}/n^{2} equals n^{3}. The quotient in simplest form is 12n^{3}. Likewise, suppose the problem were [(4m-8)/5] ÷ [(m-2)/10]. The new expression would then be [(4m-8)/5] · [10/ (m-2)]. The new numerator can be factored as [4(m-2)]10, and the new denominator can be factored as 5(m-2). Since (m-2) is a common factor for both the numerator and the denominator, (m-2)/(m-2) equals 1 and cancels out, leaving 4(10)/5, or 8.

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A rational expression has one polynomial in the numerator and one polynomial in the denominator, which means that the numerator is divided by the denominator. However, the denominator cannot be solved to zero, because division by zero is undefined.

A statement such as (x^{2} +3)/(y +6) is a rational expression, because it has one polynomial in the numerator and one polynomial in the denominator. The value of x can be set to any rational number, because any number squared will be a positive number. However, the value of y has limits. It cannot be any number that equals -6, because 6-6 equals zero. Similarly, in the rational expression 5/x, x cannot be solved by 0, as 5/0 is undefined. In the rational expression a/ (b-2), b cannot be solved by 2.

A rational expression is in the simplest form when the only common factors in the numerator and denominator are 1 and -1. Think of fractions in simplest form because their common factors have been eliminated. Expressions such as 6/8 can be simplified to ¾ because 2·3/2·4. The process is similar with rational expressions. For example, the expression (12y +24)/48y can be simplified by first factoring the numerator 12y +24 to 12(y +2). Then the new expression is [12(y +2)]/48y. Then, 12/48 can be factored to ¼. The new expression is (y +2)/4y.

Sometimes both the numerator and denominator of a rational expression need to be factored to simplest terms. Suppose the numerator is in the form 2x^{2} +x. It can be factored to simplest form as x (2x +1). Suppose the denominator is in the form 3x^{2} +2x. It can be factored to x (3x +2). The x factors cancel each other out as x/x = 1, so the new expression is (2x +1)/ (3x +2). Similarly, if the numerator is in the form (a^{2} – 1), it can be factored to (a +1) (a-1). Suppose the denominator of the expression is in the form (2a^{2}-a -1). It can be factored to (a -1) (2a +1) because a·2a equals 2a^{2}, a -2a equals –a, and (-1) (1) equals -1. The new expression is [(a+1)(a-1)]/[(a-1)( 2a+1). The (a-1)/(a-1) factor cancels out to leave a new simplified expression, (a+1)/(2a +1).

Rational expressions such as (x -3)/(3-x) are expressions where the numerator and denominator are additive inverses of each other. To simplify, multiply the numerator by -1 to change the form to -1(x-3). That flips the expression to -1(3-x)/(3-x) or -1. Interested in math tutoring services? Learn more about how we are assisting thousands of students each academic year. SchoolTutoring Academy is the premier educational services company for K-12 and college students. To learn more about how we help parents and students in Ottawa, ON, Canada: visit: Tutoring in Ottawa, ON

]]>Forms of poetry exist all over the world, some predating writing. Poems are written in specific forms from the Far East through Persia and Arabia, using specific features of the culture and language. For example, the Epic of Gilgamesh, written in Sumeria in the third millennium BCE, uses common features of Sumerian language and thought. Hebrew poetry such as the Song of Deborah dates from the 12th century BCE. It is built around a call-and-response style known as parallelism, while poetry from many lands uses acrostics and arrangement of first letters per line.

There are many different types of poetry in China. The oldest known poem book is called the Shijing (Classic of Poetry), written in many forms until 700 BCE. It contains folk songs, poetry of the Imperial Court, and songs to worship the ancestors. The Chu Ci (Songs of the South) date from 158 CE and include lyric poetry and poems about nature. Modern poetry, rather than follow strict rules of the Chinese language, is written in free verse. Like poetry in English, poetry in Chinese is written and performed in many different styles.

Japanese poetry focuses on the number of sound-units in a line rather than rhyme, using the tonal and internal features of the language. The earliest poetry followed the Chinese form, but by the 680 CE, new forms came into being. Tanka are structured in a 5-7, 5-7-7 pattern and are not usually rhymed. Haiku are shorter, in a 5-7-5 pattern, often about nature or the seasons.

The Persian language group includes the areas around Iran, Turkey, the Caucasus, western Pakistan, Afghanistan, and northern India. Some of the most famous poets include Rumi and Omar Khayyam. Rumi lived in the 13th century CE, and wrote poetry in a style called ghazal. The Rubaiyat of Omar Khayyam was translated into English in 1859. The ghazal is a type of rhyming poem that has refrains, and the Rubaiyat is arranged in units of four lines.

Arabic poetry has a long oral tradition, and classical poetry is most often rhymed. Narrative poetry, romantic poetry, and odes are common traditional forms. The One Thousand and One Nights of Scheherazade consists of interlocking stories, written in romantic verse. Modern Arabic poetry is written in many different forms and on a wider range of topics.

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]]>Poetry is based upon aesthetic and rhythmic qualities of language, and exists across all cultures of the world. Some say that poems, like music, came before literacy, and poetic tales made important facts easier to remember. There are hundreds of subtypes of poetry, and some types include narrative poetry, epic poetry, lyric poetry, and free verse.

Languages, including English, have rhythm. Poets manipulate the natural timing of their language, as expressed by stressed and unstressed syllables, to emphasize the pattern within a poem. For example, Shakespeare often wrote in a rhythm called **iambic pentameter**, which has five metric feet per line. Think of the rhythm and time signature within a song. Some music has a ¾ or 6/8 rhythm, while others have a 4/4, 7/12, or even other signatures. Similarly, poets have used different types of rhythm and meter, and some do not stick to one meter within a poem.

Some of the other ways poets use repetitive patterns is by rhyming words, either at the end of the line or in the middle, in predictable locations. Words may be hard or exact rhymes, such as **moon**, **boon**, and **noon**, or they may be soft, almost-rhyming words, such as **wing** and **caring**. Some rhymes are in the reader’s or listener’s imagination, as “This sugar is neat; it tastes so sour.” The reader expects to hear the word **sweet** and rhymes the word **sour**. (Perhaps there is lemon juice or vinegar added to the sugar, so it’s a surprise.) Other forms of repetition include alliteration, assonance, and consonance. Alliteration is the repetition of sounds at the beginning of words, such as **bike** and **book**, assonance is the repetition of vowel sounds, such as **boat** and **loan**, and consonance is the repetition of consonants such as **tell** and **wall**.

Poetry is written in many different forms. Classical sonnets have very specific requirements for rhyme and number of lines. A sonnet, by definition, has fourteen lines, and those lines rhyme in a specific pattern. Sonnets have been part of poetry from the 13^{th} century, and Shakespeare wrote many of the most famous ones. A villanelle has nineteen lines that rhyme in a specific pattern and has many refrains. Many types of free verse do not rhyme at all, and poets experiment with new forms daily. Some modern poets experiment with the forms of words to create an actual picture.

Poets have used many different genres throughout the ages. Narrative poetry passed down stories, sometimes in the form of ballads. Epic poetry told the adventures of heroes. These genres were used to pass down knowledge and belief through the generations, and some predated written language, or was a valuable tool when many people could not read. Lyric poetry, rather than tell a story, depicts inner feelings, moods, and desires.

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]]>Jellyfish are found in all the world’s oceans, from the surface to the deep sea. Most live in saltwater, although there are a few species that live in fresh water. Jellyfish have soft bodies and swim freely for much of their adult lives.

There are hundreds of different species of jellyfish. Adult jellyfish (sometimes called jellies) have an umbrella-shaped bell, with tentacles that hang from the bell. Different species of jellyfish live in every ocean, and some species live in fresh water. Some live near the surface, while others live in the deep sea. They have soft bodies, unlike the vertebrate skeletons of fish. They move through the water efficiently by rapidly expanding and contracting their bell-shaped bodies to push water behind them.

Most species of jellyfish eat plankton, small crustaceans, fish eggs, small fish, and other smaller jellyfish. Their prey are caught in the vortex created by their movement through the water, then entangled in tentacles. They are preyed upon by larger jellyfish, tuna, shark, swordfish, and sea turtles. Jellyfish tend to gather in large swarms, often called “blooms”, and some invasive species of jellyfish grow in new habitats, overwhelming other populations of marine creatures. They are relatively more adapted to survive in warmer waters that contain less oxygen but more nutrients, such as agricultural runoff.

Many species are poisonous, with toxic stingers on the tentacles. Most stings are not deadly, but a few species are highly venomous, and their stings can be fatal. The sea wasp and other forms of box jellyfish hunt their prey rather than drifting toward it. Beaches in Australia have warning signs against stinging jellyfish. The most effective protection against jellyfish stings is wearing protective covering, such as a wetsuit or even pantyhose, as the stinging cells cannot touch the skin and react with it chemically.

Eggs develop into free-swimming larvae. The free-swimming larvae settle onto firm surfaces and become polyps, similar to sea anemones. The polyp then breaks off to become an adult medusa, or jellyfish. Life spans vary depending upon the size and species. The smallest species may live only a few hours, but many large coastal species live for months. Aquarium jellyfish can live several years, as they have regular food and are tended carefully.

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