square roots of imaginary numbers

Square roots of positive and negative numbers

Square numbers and square roots of perfect squares

Decimal Approximations of Square Roots

GRADE 8 MATH CHPT SQUARE AND SQUARE ROOTS DIFFERENT TYPES OF QUESTIONS

A1.AREI.4* Solve mathematical and real-world problems involving quadratic equations in one variable. (Note: A1.AREI.4a and 4b are not Graduation Standards.) a. Use the method of completing the square to transform any quadratic equation in 𝑥 into an equation of the form (𝑥 − ℎ)2 = 𝑘 that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex

Land preparation •Carrots can be planted in beds. •The beds should be prepared before planting. (a)digging: this is done using a hoe or a spade so as to loosen the soil. •During digging , remove roots and stems of weeds. (b) manuring: add 2kg of well rotten manure per square metre. •Work the manure deeply into the soil so that the roots of the carrots do not come into directly contact with manure. •Improperly incorporated manure causes development of carrots that have side roots. •this condition is called forking. Planting •Carrots are planted in beds where they will mature. •Water the bed a day before planting. •Mark the rows 30cm apart. •Carrots seeds are very small, therefore due to their size sow the seeds thinly along the rows. •Rub the seeds between your fingers as you sow. •Cover the seeds with fine sand. •The bed can be covered with grass so that the soil remain moist. •Water the bed soon after sowing using a watering can fitted with a fine rose. Caring for carrots (a) Thinning: is when you remove some of the baby plants in a row to free up space for the rest so that they can grow to full size. • Thinning in carrots is usually done two weeks after germination. • Thin out to leave a spacing of 4 -6 cm from each plant. (b) Watering: If carrots are not watered enough they will not grow fast. • Therefore carrots need a steady supply of water to keep the soil moist during the growing period. (c) Weeding: weeds are removed by hand, especially those in the rows. (d) Top dressing: carrots are usually not top dressed. •Fertilizer can only be applied after one has noticed leaves turning yellow. (e) Pest and disease control: aphids and nematodes attack carrots. •Dimethoate is used to control aphids. Harvesting carrots •Carrots are harvested 2 to 3 months after planting. •Bigger ones are harvested first inorder to leave more room for the young seedlings to develop. •Water the bed first the lift the carrots by hand. •A garden fork can also be used to loosen the soil, then pull up by hand. •The carrots can then be tied into bunches of 5 – 7 carrots

Figure 18-11 represents the amount of energy stored as organic material in each trophic level in an ecosystem. The pyramid shape of the diagram indicates the low percentage of energy transfer from one level to the next. On average, 10 percent of the total energy consumed in one trophic level is incor- porated into the organisms in the next. Why is the percentage of energy transfer so low? One reason is that some of the organisms in a trophic level escape being eaten. They eventually die and become food for decomposers, but the energy contained in their bodies does not pass to a higher trophic level. Even when an organism is eaten, some of the molecules in its body will be in a form that the consumer cannot break down and use. For example, a cougar cannot extract energy from the antlers, hooves, and hair of a deer. Also, the energy used by prey for cellu- lar respiration cannot be used by predators to synthesize new bio- mass. Finally, no transformation or transfer of energy is 100 percent efficient. Every time energy is transformed, such as during the reactions of metabolism, some energy is lost as heat. Limitations of Trophic Levels The low rate of energy transfer between trophic levels explains why ecosystems rarely contain more than a few trophic levels. Because only about 10 percent of the energy available at one trophic level is transferred to the next trophic level, there is not enough energy in the top trophic level to support more levels. Organisms at the lowest trophic level are usually much more abundant than organisms at the highest level. In Africa, for exam- ple, you will see about 1,000 zebras, gazelles, and other herbivores for every lion or leopard you see, and there are far more grasses and shrubs than there are herbivores. Higher trophic levels con- tain less energy, so, they can support fewer individuals.A population is a group of organisms that belong to the same species and live in a particular place at the same time. All of the bass living in a pond during a certain period of time make up a pop- ulation because they are isolated in the pond and do not interact with bass living in other ponds. The boundaries of a population may be imposed by a feature of the environment, such as a lake shore, or they can be arbitrarily chosen to simplify a study of the population. The humans shown in Figure 19-1 are part of the pop- ulation of a city. The properties of populations differ from those of individuals. An individual may be born, it may reproduce, or it may die. A population study focuses on a population as a whole—how many individuals are born, how many die, and so on. Population Size A population’s size is the number of individuals that the population contains. Size is a fundamental and important population property but can be difficult to measure directly. If a population is small and composed of immobile organisms, such as plants, its size can be determined simply by counting individuals. Often, though, individ- uals are too abundant, too widespread, or too mobile to be counted easily, and scientists must estimate the number of individuals in the population. Suppose that a scientist wants to know how many oak trees live in a 10 km2 patch of forest. Instead of searching the entire patch of forest and counting all the oak trees, the scientist could count the trees in a smaller section of the forest, such as a 1 km2 area. The scientist could then use this value to estimate the population of the larger area. SECTION 1 OBJECTIVES ● Describe the main properties that scientists measure when they study populations. ● Compare the three general patterns of population dispersion. ● Identify the measurements used to describe changing populations. ● Compare the three general types of survivorship curves. VOCABULARY population population density dispersion birth rate death rate life expectancy age structure survivorship curve FIGURE 19-1 A population can be widely distributed, as Earth’s human population is, or confined to a small area, as species of fish in a lake are. Copyright © by Holt, Rinehart and Winston. All rights reserved. 382 CHAPTER 19 If the small patch contains 25 oaks, an area 10 times larger would likely contain 10 times as many oak trees. A similar kind of sampling technique might be used to estimate the size of the pop- ulation shown in Figure 19-2. To use this kind of estimate, the sci- entist must assume that the distribution of individuals in the entire population is the same as that in the sampled group. Estimates of population size are based on many such assumptions, so all esti- mates have the potential for error. Population Density Population density measures how crowded a population is. This measurement is always expressed as the number of individuals per unit of area or volume. For example, the population density of humans in the United States is about 30 people per square kilome- ter. Table 19-1 shows the population sizes and densities of humans in several countries in 2003. These estimates are calculated for the total land area. Some areas of a country may be sparsely popu- lated, while other areas are very densely populated. Dispersion A third population property is dispersion (di-SPUHR-zhuhn). Dispersion is the spatial distribution of individuals within the popu- lation. In a clumped distribution, individuals are clustered together. In a uniform distribution, individuals are separated by a fairly con- sistent distance. In a random distribution, each individual’s location is independent of the locations of other individuals in the popula- tion. Figure 19-3 illustrates the three possible patterns of dispersion. Clumped distributions often occur when resources such as food or living space are clumped. Clumped distributions may also occur because of a species’ social behavior, such as when animals gather into herds or flocks. Uniform distributions may result from social behavior in which individuals within the same habitat stay as far away from each other as possible. For example, a bird may locate its nest so as to maximize the distance from the nests of other birds. These migrating wildebeests in East Africa are too numerous and mobile to be counted. Scientists must use sampling methods at several locations to monitor changes in the population size of the animals. FIGURE 19-2 TABLE 19-1 Population Size and Density of Some Countries Population size Population density Country (in millions) (in individuals/km2) China 1,289 135 India 1,069 325 United States 292 30 Russia 146 8 Japan 128 337 Mexico 105 54 Kenya 32 54 Australia 20 3 dispersion from the Latin dis-, meaning “out,” and spargere, meaning “to scatter” Word Roots and Origins Copyright © by Holt, Rinehart and Winston. All rights reserved. POPULATIONS 383 The social interactions of birds called gannets, which are shown in Figure 19-3b, result in a uniform distribution. Each gannet chooses a small nesting area on the coast and defends it from other gannets. In this way, each gannet tries to maximize its distance from all of its neighbors, which causes a uniform distribution of individuals. Few populations are truly randomly dispersed. Rather, they show degrees of clumping or uniformity. The dispersion pattern of a population sometimes depends on the scale at which the popu- lation is observed. The gannets shown in Figure 19-3b are uni- formly distributed on a scale of a few meters. However, if the entire island on which the gannets live is observed, the distribution appears clumped because the birds live only near the shore. POPULATION DYNAMICS All populations are dynamic—they change in size and composition over time. To understand these changes, scientists must know more than the population’s size, density, and dispersion. One important measure is the birth rate, the number of births occur- ring in a period of time. In the United States, for example, there are about 4 million births per year. A second important measure is the death rate, or mortality rate, which is the number of deaths in a

Southeast Asia, vast region of Asia situated east of the Indian subcontinent and south of China. It consists of two dissimilar portions: a continental projection (commonly called mainland Southeast Asia) and a string of archipelagoes to the south and east of the mainland (insular Southeast Asia). Extending some 700 miles (1,100 km) southward from the mainland into insular Southeast Asia is the Malay Peninsula; this peninsula structurally is part of the mainland, but it also shares many ecological and cultural affinities with the surrounding islands and thus functions as a bridge between the two regions. Mainland Southeast Asia is divided into the countries of Cambodia, Laos, Myanmar (Burma), Thailand, Vietnam, and the small city-state of Singapore at the southern tip of the Malay Peninsula; Cambodia, Laos, and Vietnam, which occupy the eastern portion of the mainland, often are collectively called the Indochinese Peninsula. Malaysia is both mainland and insular, with a western portion on the Malay Peninsula and an eastern part on the island of Borneo. Except for the small sultanate of Brunei (also on Borneo), the remainder of insular Southeast Asia consists of the archipelagic nations of Indonesia and the Philippines. Southeast Asia stretches some 4,000 miles at its greatest extent (roughly from northwest to southeast) and encompasses some 5,000,000 square miles (13,000,000 square km) of land and sea, of which about 1,736,000 square miles is land. Mount Hkakabo in northern Myanmar on the border with China, at 19,295 feet (5,881 meters), is the highest peak of mainland Southeast Asia. Although the modern nations of the region are sometimes thought of as being small, they are—with the exceptions of Singapore and Brunei—comparatively large. Indonesia, for example, is more than 3,000 miles from west to east (exceeding the west-east extent of the continental United States) and more than 1,000 miles from north to south; the area of Laos is only slightly smaller than that of the United Kingdom; and Myanmar is considerably larger than France. All of Southeast Asia falls within the tropical and subtropical climatic zones, and much of it receives considerable annual precipitation. It is subject to an extensive and regular monsoonal weather system (i.e., one in which the prevailing winds reverse direction every six months) that produces marked wet and dry periods in most of the region. Southeast Asia’s landscape is characterized by three intermingled physical elements: mountain ranges, plains and plateaus, and water in the form of both shallow seas and extensive drainage systems. Of these, the rivers probably have been of the greatest historical and cultural significance, for waterways have decisively shaped forms of settlement and agriculture, determined fundamental political and economic patterns, and helped define the nature of Southeast Asians’ worldview and distinctive cultural syncretism. It also has been of great importance that Southeast Asia, which is the most easily accessible tropical region in the world, lies strategically astride the sea passage between East Asia and the Middle Eastern–Mediterranean world. Within this broad outline, Southeast Asia is perhaps the most diverse region on Earth. The number of large and small ecological niches is more than matched by a staggering variety of economic, social, and cultural niches Southeast Asians have developed for themselves; hundreds of ethnic groups and languages have been identified. Under these circumstances, it often is difficult to keep in mind the region’s underlying unity, and it is understandable that Southeast Asia should so often be treated as a miscellaneous collection of cultures that simply do not quite fit anywhere else. Roofs of the Forbidden City, Beijing, China Britannica Quiz All About Asia Yet from ancient times Southeast Asia has been considered by its neighbors to be a region in its own right and not merely an extension of their own lands. The Chinese called it Nanyang and the Japanese Nan’yō, both names meaning “South Seas,” and South Asians used such terms as Suvarnabhūmi (Sanskrit: “Land of Gold”) to describe the area. Modern scholarship increasingly has yielded evidence of broad commonalities uniting the peoples of the region across time. Studies in historical linguistics, for example, have suggested that the vast majority of Southeast Asian languages—even many of those previously considered to have separate origins—either sprang from common roots or have been long and inseparably intertwined. Despite inevitable variation among societies, common views of gender, family structure, and social hierarchy and mobility may be discerned throughout mainland and insular Southeast Asia, and a broadly common commercial and cultural inheritance has continued to affect the entire region for several millennia. These and other commonalities have yet to produce a conscious or precise Southeast Asian identity, but they have given substance to the idea of Southeast Asia as a definable world region and have provided a framework for the comparative study of its components.

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