Transcription of Basic Algebra Rules - UCSD Mathematics
1 Basic Algebra Leta,b,c, anddbe numbers.(a)You can break up a fraction from a sum in the numerator, butnotin the denom-inator:a+bc=ac+bcbutab+c6=ab+ac(b) Cancellation of thechere requires that it appears ineachadditive term of thenumerator and denominator:ca+cbcd=c(a+b)cd=a+bdbutca+b cd6=a+bd(c)Compound fractions can be simplified by using the rule division is the same asmultiplication by the reciprocal :abcd=ab cd=ab dc= Logs. Letaandbbe numbers.(a)Natural logs distribute in a funny way over products and quotients:ln (ab) = lna+ lnbln (ab) = lna lnbbut they donotdistribute over sums:lna+b6= lna+ lnb(b)Natural logs can help you work with exponents by bringing them down :ln (ab) = Leta,b,m, andnbe numbers.(a)Exponents distribute over products, butnotover sums:(ab)n=anbnbut(a+b)n6=an+bn(b)A negative exponent can always be viewed as a denominator, and vice versa:a n=1an(c)Two terms with exponents can only be multiplied if they share the same base; inthat case, the exponents add:aman=am+nbutamdncannot be further simplified, andaman6=amn(d)Similarly for division:aman=am Leta,b,m, andnbe numbers.
2 (a)Remember that roots can always be viewed as fractional exponents:n a=a1nWith this point of view, we ll inherit all the Rules about exponents. In particular,(b)Distributing a root over a product:n ab= (ab)1n=a1nb1n=n an b(c)Multiplying two roots with a common base:m an a=a1ma1n=a1m+1n2
