Tiêu đề Math-Readings-2.pdf ✅

Academic- Academic-Clinic.com Clinic.com Clinic.com The achiever’s guide to academic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Math II - 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-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! INTERMEDIATE ALGEBRA POINTERS Powers and roots Operations with powers. 1. At multiplying of powers with the same base their exponents are added: a m · a n = a m + n . 2. At dividing of powers with the same base their exponents are subtracted: 3. A power of product of two or some factors is equal to a product of powers of these factors: ( abc... ) n = a n · b n · c n ... 4. A power of a quotient (fraction) is equal to a quotient of powers of a dividend (numerator) and a divisor (denominator): ( a / b ) n = a n / b n . 5. At raising of a power to a power their exponents are multiplied: ( a m ) n = a m n . All above mentioned formulas are read and executed in both directions – from the left to the right and back. E x a m p l e . ( 2 · 3 · 5 / 15 ) 2 = 2 2 · 3 2 · 5 2 / 15 2 = 900 / 225 = 4 . Operations with roots. In all below mentioned formulas a symbol means an arithmetical root ( all radicands are considered here only positive ). 1. A root of product of some factors is equal to a product of roots of these factors:
Academic- Academic-Clinic.com Clinic.com Clinic.com The achiever’s guide to academic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Math II - 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-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! 2. A root of a quotient is equal to a quotient of roots of a dividend and a divisor: 3. At raising a root to a power it is sufficient to raise a radicand to this power: 4. If to increase a degree of a root by n times and to raise simultaneously its radicand to the n-th power, the root value doesn’t change: 5. If to decrease a degree of a root by n times and to extract simultaneously the n-th degree root of the radicand, the root value doesn’t change: Widening of the power notion. Till now we considered only natural exponents of powers; but operations with powers and roots can result also to negative, zero and fractional exponents. All these exponents of powers require to be defined. Negative exponent of a power. A power of some number with a negative (integer) exponent is defined as unit divided by the power of the same number with the exponent equal to an absolute value of the negative exponent: Now the formula a m : a n = a m - n may be used not only if m is more than n , but also for a case if m is less than n . n n n a /b = a / b ( ) n m m n a = a
Academic- Academic-Clinic.com Clinic.com Clinic.com The achiever’s guide to academic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Math II - 3 - 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! E x a m p l e . a 4 : a 7 = a 4 - 7 = a -3 . If we want the formula a m /a n = a m - n to be valid at m = n we need the definition of zero exponent of a power. Zero exponent of a power. A power of any non-zero number with zero exponent is equal to 1. E x a m p l e s: 2 0 = 1, ( – 5 ) 0 = 1, ( – 3 / 5 ) 0 = 1. Fractional exponent of a power. To raise a real number a to a power with an exponent m / n it is necessary to extract the n-th degree root from the m-th power of this number a: E x a m p l e : About meaningless expressions. There are some expressions: Case 1. where a ≠ 0 , doesn’t exist. Really, if to assume that where x – some number, then according to the definition of a division we have: a = 0 · x , i.e. a = 0 , but this result contradicts to the condition: a ≠ 0 . Case 2. is any number. Really, if to assume that this expression is equal to some number x, then according to the definition of a division: 0 = 0 · x . But this equality is valid at any number x, which was to be proved. Case 3. If to assume, that rules of operations with powers are spread to powers with a zero base, then n n m m a = a 4 4 64 8 2 3 3 = = =
Academic- Academic-Clinic.com Clinic.com Clinic.com The achiever’s guide to academic life and beyond... Website: http://academic-clinic.com Facebook: http://www.facebook.com/academic.clinic Twitter: http://twitter.com/acadclinic College entrance exam and science high school entrance test tips. Conquer UPCAT, ACET, USTET, DLSUCET, PSHS-NCE, and other entrance tests. Math II - 4 - 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! 0 0 is any number . S o l u t i o n . Consider the three main cases: 1) x = 0 – this value doesn’t satisfy the equation ( Why ? ) ; 2) at x > 0 we receive: x / x = 1, i.e.1 = 1, hence, x – any number, but taking into consideration that in this case x > 0, the answer is: x > 0 ; 3) at x < 0 we receive: – x / x = 1, i.e. –1 = 1, and the answer is: there is no solution in this case. So, the answer: x > 0 . Formulas of abridged multiplication From the rules of multiplication of sums and polynomials the following seven formulas of abridged multiplication can be easily received. It is necessary to know them by heart, as they are used in most of problems in mathematics. [1] ( a + b )2 = a2 + 2ab + b2 , [2] ( a – b )2 = a2 – 2ab + b2 , [3] ( a + b ) ( a – b ) = a2 – b2, [4] ( a + b )3 = a3 + 3a2 b + 3ab2 + b3 , [5] ( a – b )3 = a 3 – 3a2 b + 3ab2 – b3 , [6] ( a + b )( a2 – ab + b2 ) = a3 + b3 ,