Mathematics reading reviewer, Study Guides, Projects, Research of Mathematics

Download Mathematics reading reviewer and more Study Guides, Projects, Research Mathematics in PDF only on Docsity! + Academic-Clinic.com The achiever's guide to acadewic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic eS College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. BASIC ARITHMETIC AND ALGEBRA POINTERS Whole (natural) numbers Natural numbers - numbers, which appear as a result of calculus of single subjects: peoples, animals, birds, trees, different wares and so on. Series of natural numbers: 1, 2, 3, 4, 5, ... is continued endlessly and is called natural series. Arithmetical operations Addition — an operation of finding a sum of some numbers: 11 + 6 = 17. Here 11 and 6 — addends, 17 = the sum. If addends are changed by places, a sum is saved the same: 11 + 6 = 17 and 6 +11 = 17. Subtraction — an operation of finding an addend by a sum and another addend: 17 — 6 = 11. Here 17 is a minuend, 6—a subtrahend, 11 — the difference. Multiplication. To multiply one number n (a multiplicand ) by another m (a multiplier ) means to repeat a multiplicand nas an addend m times. The result of multiplying is called a product. The operation of multiplication is written as: n x m or n- m. For example, 12 x 4 = 12 + 12 + 12 + 12 = 48. In our case 12 x 4= 48 or 12 4 = 48. Here 12 is a multiplicand, 4 — a multiplier, 48 — a product. If a multiplicand n and a multiplier m are changed by places, their product is saved the same: 12-4 = 12+ 12+12+12==48 and 4-12=4+ 44+44+44+44+4+4+4+4+4+4+44 = 48. Therefore, a multiplicand and a multiplier are called usually factors or multipliers. Division — an operation of finding one of factors by a product and another factor: 48 : 4 = 12. Here 48 is a dividend, 4 — a divisor, 12 — the quotient. At dividing integers a quotient can be not a whole number. Then this quotient can be present as a fraction. If a quotient is a whole number, then it is called that numbers are divisible, ie. one number is divided without remainder by another. Otherwise, we have a division with remainder. For example, 23 isn’t divided by 4 ; this case can be written as: 23=5- 4+ 3. Here 3 is a remainder. Raising to a power. To raise a number to a whole (second, third, forth, fifth etc.) power means to repeat it as a factor two, three, four, five and so on. The number, repeated as a factor, is called a base of a power; the quantity of factors is called an index or an exponent of a power; the result is called a value of a power. A raising to a power is written as: Math I -1- Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clin: Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! + Academic-Clinic.com The achiever's guide to acadewic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic eS College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. 3° =3-3-3.3-3=243. Here 3 — a base of the power, 5 — an exponent (an index) of the power, 243 — a value of the power. The second power is called a square, the third one — a cube. The first power of any number is this number. Extraction of a root — an operation of finding a base of a power by the power and its exponent: 3 243 =3. Here 243 - a radicand, 5 — an index (degree) of the root, 3 — a value of the root. The second root is called a square root, the third root — a cube root.The second degree of square root isn’t written: ¥16 =4 Addition and subtraction, multiplication and division, raising to a power and extraction of a root are two by two. mutually inverse operations. Order of operations. Brackets If brackets are absent, the following order of operations is right: 1) ng to a power and extraction of a root (one after another); 2) multiplication and division (one after another); 3) addition and subtraction (one after another). If brackets are present, at first all operations inside brackets are executed according to the aforesaid order, and then the rest of the operations out of brackets are executed (in the same order). Example. Calculate the next expression: (10+2°-3)+4° - (16:2 -1)-5-150:5°. Solution. At first, powers must be calculated and changed by theirs values: (10 + 8- 3) +64- (16: 2-1) - 5-150: 25; Math I -2- Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clin: Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! ACademic-Clinic.com The achiever's guide to acadewic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic eS College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. 3) to write out all common factors in these factorizations; 4) to take the least power of each of them, meeting in the all factorizations; 5) to multiply these powers. Example. Find GCF for numbers: 168, 180 and 3024. Solution. 168=2-2-2-3-7=2) -3!.7', 180=2-2-3-3-5=2? .3?.5', 3024=2-2-2-2-3-3-3-7=24.3-7!. Write out the least powers of the common factors 2 and 3 and multiply them: GCF=2° -3' =12. Least common multiple Common multiple of some numbers is called a number, which is divisible by each of them. For example, numbers 9, 18 and 45 have as a common multiple 180. But 90 and 360 are also theirs common multiples. Among all common multiples there is always the least one, in our case this is 90. This number is called a least common multiple (LCM). To find a least common multiple (LCM) of some numbers it is necessary: 1) to express each of the numbers as a product of its prime factors, for example: 504=2-2-2-3-3-7, 2) to write powers of all prime factors in the factorization a 504=2-2-2-3-3-7=2) 3? -7!, 3) to write out all prime factors, presented at least in one of these numbers; 4) to take the greatest power of each of them, meeting in the factorizations; 5) to multiply these powers. Math I -5- Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clin: Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! ACademic-Clinic.com The achiever's guide to acadewic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic eS College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Example. Find LCM for numbers: 168, 180 and 3024. Solution. 168=2-2-2-3-7=2) -3'-7', 180=2-2-3-3-5=2 St, 3024=2-2-2-2-3-3-3-7=2*-3°-7!. Write out the greatest powers of all prime factors: 2°, 3°, 5', 7! and multiply them: LCM =2*-3°-5-7=15120. Divisibility criteria Divisibility by 2. A number is divisible by 2, if its last digit is 0 or is divisible by 2. Numbers, which are divisible by 2 are called even numbers. Otherwise, numbers are called odd numbers. Divisibility by 4. A number is divisible by 4, if its two last digits are zeros or they make a two-digit number, which is divisible by 4. Divisibility by 8. A number is divisible by 8, if its three last digits are zeros or they make a three-digit number, which is divisible by 8. Divisibility by 3 and by 9 . A number is divisible by 3, if a sum of its digits is divisible by 3. A number is divisible by 9, if a sum of its digits is divisible by 9. Divisibility by 6. A number is divisible by 6, if it is divisible by 2 and by 3. Divisibility by 5. A number is divisible by 5, if its last digit is 0 or 5. Divisibility by 25. A number is divisible by 25, if its two last digits are zeros or they make a number, which is divisible by 25. Divisibility by 10. A number is divisible by 10, if its last digit is 0. Math I -6- Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clinic’s Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! + Academic-Clinic.com The achiever's guide to acadewic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic eS College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Divisibility by 100. A number is divisible by 100, if its two last digits are zeros. Divisibility by 1000. A number is divisible by 1000, if its three last digits are zeros. Divisibility by 11. A number is divisible by 11 if and only if sum of its digits, located on even places is equal to asum of its digits, located on odd places, OR these sums are differed by a number, which is divisible by 11. There are criteria of divisibility for some other numbers, but these criteria are more difficult and not considered in a secondary school program. Example. A number 378015 is divisible by 3, because a sum of its digits 3 +7+8+0+1+5=24, which is divisible by 3. This number is divisible by 5, because its last digit is 5. At last, this number is divisible by 1 1, because a sum of even digits: 7 + 0 + 5 =12 and a sum of odd digits: 3 + 8 + 1 = 12 are equal. But this number isn’t divisible by 2, 4, 6, 8,9, 10, 25, 100 and 1000, because ... Check these cases yourself ! Simple fractions A part of a unit or some equal parts of a unit is called a vulgar (simple) fraction. A number of equal parts into which a unit has been divided, is called a denominator; a number of these taken parts, is called a numerator. A fraction record: —- or BIT Here 3 — anumerator, 7 — a denominator. If a numerator is less than a denominator, then the fraction is less than | and called a proper fraction. If a numerator is equal to a denominator, the fraction is equal to 1. If a numerator is greater than a denominator, the fraction is greater than 1. In both last cases the fraction is called an improper fraction. If a numerator is divisible by a denominator, then this fraction is equal to a quotient: 63/7 = 9. Ifa division is executed with a remainder, then this improper fraction can be presented as a mixed number: 65 2 = 9 ? 7 Math I -7- Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clin: Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! + Academic-Clinic.com The achiever's guide to acadewic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic eS College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Addition and subtraction of fractions. If denominators of fractions are the same, then in order to add the fractions it is necessary to add their numerators; in order to subtract the fractions it is necessary to subtract their numerators (in the same order). The received sum or difference will be a numerator of the result; a denominator is saved the same. If denominators of fractions are different, before these operations it is necessary to reduce fractions to a common denominator. At addition of mixed numbers a sum of integer parts and a sum of fractional parts are found separately. At subtracting mixed numbers we recommend at first to reduce the mixed numbers to improper fractions, then to subtract these fractions and after this to convert the result into a mixed number again (in case of need). 2 23 14 BF 56 31 ? 4 --=—— - —=-- - -- = = 2 3 4 3 12 12°12 12 Muttiplication of fractions. To multiply some number by a fraction means to multiply it by a numerator and to divide a product by a denominator. Hence, we have the general rule for multiplication of fractions: to multiply one fraction by another it is necessary to multiply separately their numerators and denominators and to divide the first product by the second. Example. 2 5 Division of fractions. To divide some number by a fraction it is necessary to multiply this number by a reciprocal fraction. This rule follows from the definition of division (see the section “Arithmetical operations”). 3 12 3 25 3-25 5 Example. = = = 5 25 5 12 5-12 4 Decimal fractions (decimals) Decimal fraction is a result of dividing of unit by ten, hundred, thousand parts etc. These fractions are very comfortable in calculations, because they are based on the same system, that calculus and record of integers are built. Due to this both record and rules of operations with decimal fractions are actually the same as for integers. Math I -10- Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clinic’s Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our w 5 0 4 7 9.5047 = 9+ --+ + + For UPCAT, ACET, DLS’ 10 100 1000 10000 her college entrance exam information, visit the Acauenne-Caime wevsite. 1e your menus and classmates to come find and join us. The more, the merrier. Good luck! ACademic-Clinic.com The achiever's guide to acadewic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic eS College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. At recording decimal fractions it isn’t necessary to mark parts (as denominator ); this is known by place, that the corresponding digit occupies. At first the integer part of a number is written; to the right of it the decimal point is put; the first digit after the point means a number of tenths (a number of tenth parts of unit ), the second — a number of hundredths, the third — thousandths, and so on. Digits, located after decimal point, are called decimal places. Example. One of advantages of decimals — they are easily reduced to the shape of vulgar fractions: a number after a decimal point (5047 in our case ) is a numerator, and the n-th power of 10 (n— a quantity of decimal places, in our case n= 4 ) is a denominator: 5047 9.5047= 9 ——— 10000 Ifa decimal doesn’t contain an integer part, zero is put before a decimal point: 13 —-=013 100 Properties of decimals. 1. A decimal fraction isn't changed, if to add some zeros to the right of it: 13.6 =13.6000. 2. A decimal fraction isn’t changed, if to reject zeros, located in the end: Math I -11- Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clin: Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! ACademic-Clinic.com The achiever's guide to acadewic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic eS College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. 0.00123000 = 0.00123 Note: it’s prohibited to reject zeros, located not in the end of a decimal! 3. A decimal fraction will be increased by 10, 100, 1000 ,...times, if to transfer a decimal point to one, two, three, ... places to the right: 3.675 ---> 367.5 (it increases by 100 times). 4. A decimal fraction will be decreased by 10, 100, 1000.,...times, if to transfer a decimal point to one, two, three, ... places to the left: 1536.78 ---> 1.53678 (it decreases by 1000 times) These properties permit quickly to multiply and to divide decimal fractions by 10, 100, 1000 and so on. Repeating decimal is a decimal in which a digit or a group of digits repeats endlessly in a pattern. This group of repeating digits is called a period of decimal and is written in brackets. For instance, Example. If to divide 47 by 11, then the result is 4.27272727... = 4.(27) Operations with decimal fractions Addition and subtraction of decimals, These operations are executed as well as an addition and a subtraction of whole numbers. It is only necessary to write the corresponding decimal places one under another. Example. 307 + 11.354 0.008 14.433 Math I -12- Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clinic’s Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! + Academic-Clinic.com The achiever's guide to acadewic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic eS College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Example. Convert 5/8 toa decimal fraction. Solution. Dividing 5 by 8, we'll receive 0.625. ( Check it, please ! ) can be continued infinitely. Then a simple fraction cannot be converted exactly is never required. Dividing is broken if decimal places, that are of interest, In the most of cases this proce: to adecimal. But in practice tl have been already received. Example.Convert 1/3 to a decimal fraction. Solution. Dividing | by 3 will be infinite: 1:3 = 0.3333... . Check it, please. Percents Percent is a hundredth part of unit. A record 1% means 0.01. There are three main problems by percents: Problem 1. Find an indicated percent ofa given number. The given number is multiplied by the indicated number of percents; then a product is divided by 100. Example. A deposit in a bank has an annual increase 6%. A sum of money in the beginning was equal to $10000. How many dollars will the sum be increased by in the end of the year? Solution: $10000 - 6/100 = $600. Problem 2. Find a number by another given number and its percent value of the unknown number. The given number is divided by its percent value; the result is multiplied by 100. Example. — A salary by January was equal to $15000, that was equal 7.5% of an annual salary. What was the annual salary ? Solution: $15000/7.5 - 100 =$200000. Problem 3. Find the percent expression of one number by another. ‘The first number is divided by the second, and a result is multiplied by 100. Example. 0n2001 aplant have produced 40000 cars; and on 2002 - only 36000 cars, What percent Math I -15- Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clin: Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! + Academic-Clinic.com The achiever's guide to acadewic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic eS College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. does it constitute relatively to the output of 2001 ? Solution: 36000/ 40000 - 100 = 90%. Ratio and proportion. Proportionality Ratio is a quotient of dividing one number by another. Proportion — an equality of two ratios. For instance: 12:20=3:5; a:b=c:d. Border terms of the proportion: 12 and 5 in the first proportion; aand din the second proportion. Middle terms of the proportion: 20 and 3 in the first proportion; band c in the second proportion. The main property of a proportion: A product of border terms of a proportion is equal to a product of its middle terms. Two mutually dependent values are called proportional ones, if a ratio of their values is saved as invariable. This invariable ratio of proportional values is called a factor of a proportionality. Example. A mass of any substance is proportional to its volume. For instance, 2 liters of mercury weigh 27.2 kg, 5 liters weigh 68 kg, 7 liters weigh 95.2 kg. A ratio of mercury mass to its volume ( factor of a proportionality ) will be equal to: 272 68 952 = ——-= ——--= 13.6 — amercury density 2 3 7 Thus, a factor of a proportionality in this example is density. Math I -16- Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clin: Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! + Academic-Clinic.com The achiever's guide to acadewic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic eS College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. The Integers and Rational Numbers To the natural numbers one adjoins their negatives and zero to form the integers. The ratios a/b of the integers, where a and b are integers and b /= 0, constitute the rational numbers; the integers are those rational numbers for which b = 1. The rational numbers may also be represented by repeating decimals; e.g., 1/2 = 0.5000 ..., 2/3 = 0.6666 ..., 2/7 = 0.2857 14285714 ... Negative integers appear, when the greater integer is subtracted from the smaller one, for instance: 10-15 =-5 The sign “minus” before 5 shows, that this number is negative. Series of negative integers continue endles -l, -2,-3,-4,-5... Fractional negative numbers appear, for example, when the greater number is subtracted from the smaller one: 3 12 3 Also it is possible to say, that fractional negative numbers appear as a result division of a negative integer by a natural number: 13 “13472 -— 7 Positive numbers in contrast to negative numbers (integers and fractional ones), are the numbers, considered in arithmetic (also integers and fractional ones). Rational numbers — positive and negative numbers (integers and fractional ones) and zero. The more exact definition of rational numbers, adopted in mathematics, is the following: Math I -17- Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clin: Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! + Academic-Clinic.com The achiever's guide to acadewic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic eS College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. where a, b — real numbers, i — an imaginary unit. In more details about complex numbers see the section “Complex numbers”. Examples ofcomplex numbers: 344i, 7-13.61, 0+25i1=25i, 247i. Monomials and polynomials Monomial is a product of two or some factors, each of them is either a number, or a letter, or a power of a letter. For example, 3a°b*, bd*®, -I7abe are monomials. A single number or a single letter may be also considered as a monomial. Any factor of a monomial may be called a coefficient. Often only a numerical factor is called a coefficient. Monomials are called similar or like ones, if they are identical or differed only by coefficients. Therefore, if two or some monomials have identical letters or their powers, they are also similar (like) ones. Degree of monomial is a sum of exponents of the powers of all its letters. Addition of monomials. If among a sum of monomials there are similar ones, he sum can be reduced to the more simple form: ax?y? -Sb3x3y*4e%x3y?=(a-Sbite%)xiy?. This operation is called reducing of like terms. Operation, done here, is called also taking out of brackets. Multiplication of monomials, A product of some monomials can be simplified, only if it has powers of the same letters or numerical coefficients. In this case exponents of the powers are added and numerical coefficients are multiplied. Example: 5 ax>z*(-7a>x3y?)= -35aix% yz. Math I = 20- Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clin: Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! + Academic-Clinic.com The achiever's guide to acadewic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic eS College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Division of monomials. A quotient of two monomials can be simplified, if a dividend and a divisor have some powers of the same letters or numerical coefficients. In this case an exponent of the power in a divisor is subtracted from an exponent of the power in a dividend; a numerical coefficient of a dividend is divided by a numerical coefficient of a divisor. Example: 35 a*x?z°:7ax>z°=Sa°xz’. Polynomial is an algebraic sum of monomials. Degree of polynomial is the most of degrees of monomials, forming this polynomial. Multiplication of sums and polynomials: a product of the sum of two or some expressions by any expression is equal to the sum of the products of each of the addends by this expression: (p+ q+r)a=pat gat ra - opening of brackets. Instead of the letters p, g, r, a any expressions can be taken. Example: (x+ y+ z at b )=x( at b )+ y(atb) + eatb)= =xa+xb+ya+yb+ zat zb. A product of sums is equal to the sum of all possible products of each addend of one sum to each addend of the other sum. Algebraic fractions Algebraic fraction is an expression of a shape A/ B, where A and B can be a number, a monomial, a polynomial. As in arithmetic, A is called a numerator, B — a denominator. Arithmetical fraction is a particular case of an algebraic one. Canceling fractions Example: 3a?-Sab+262 (Za<Tb)(a-b) ab 3a? + ab — 2b? (BexTh (ath) ate wo1- Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clin: Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck! ACademic-Clinic.com The achiever's guide to acadewic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic eS College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Addition and subtraction of fractions To add or to subtract two or some fractions it is necessary to make the same operations as in arithmetic. Example: a b ax +be + = eka ex? ctx? Multiplication and division of fractions Multiplication and division of algebraic fractions doesn’t differ from the same operations in arithmetic. Canceling a fraction can be done both before and after multiplication of numerators and denominators. Example: Zab? 4a*h Zab? . Ze Shy oxy Says omy? - 4aeb 2ax Math I = 22- Feel free to pass this on to your friends, but please don’t post it online. Discuss UPCAT and other college entrance exam questions and answers at Academic-Clinic’s Facebook Page. We encourage you to answer the questions we post there and actively participate in the discussions on our wall. For UPCAT, ACET, DLSUCET and USTET tips, tricks, news and other college entrance exam information, visit the Academic-Clinic website. Tell your friends and classmates to come find and join us. The more, the merrier. Good luck!