Quantum Physics

A.4 Appendix A Mathematical ReviewTABLE A.1Rules of Exponentsx 0 1x 1 xx n x m x nmx n /x m x nmx 1/n n √x(x n ) m x nmWhen dividing the powers of a given quantity, note thatFor example, x 8 /x 2 x 8 2 x 6 .A power that is a fraction, such asx nx m xnm13 ,corresponds to a root as follows:[A.4]x 1/n n √x[A.5]For example, 4 1/3 √ 3 4 1.587 4. (A scientific calculator is useful for suchcalculations.)Finally, any quantity x n that is raised to the mth power is(x n ) m x nm [A.6]Table A.1 summarizes the rules of exponents.EXERCISESVerify the following:1. 3 2 3 3 2432. x 5 x 8 x 33. x 10 /x 5 x 154. 5 1/3 1.709 975 (Use your calculator.)5. 60 1/4 2.783 158 (Use your calculator.)6. (x 4 ) 3 x 12C. FactoringSome useful formulas for factoring an equation areax ay az a(x y z) common factora 2 2ab b 2 (a b) 2 perfect squarea 2 b 2 (a b)(a b)differences of squaresD. Quadratic EquationsThe general form of a quadratic equation isax 2 bx c 0[A.7]where x is the unknown quantity and a, b, and c are numerical factors referred toas coefficients of the equation. This equation has two roots, given byx b √b 2 4ac[A.8]2aIf b 2 4ac, the roots will be real.EXAMPLEThe equation x 2 5x 4 0 has the following roots corresponding to the two signs of the square root term:x 5 √52 (4)(1)(4)2(1)5 √92 5 32that is,x 5 321x 5 32where x refers to the root corresponding to the positive sign and x refers to the root corresponding to thenegative sign.4