How To Add Fractions That Don't Have A Common Denominator

find the sum. express the answer as asimplified rational expression, and statethe domains. we have these two rationalexpressions, or two fractions, if you will.

How To Add Fractions That Don't Have A Common Denominator, and whenever we add fractions,we need to find a common denominator, and the commondenominator has to be something that's divisible byboth of these denominators. in general, we want to findthe least common, or the

smallest, multiple of thesenumbers, or the smallest number that's divisibleby both. when you look at it immediately,it might pop out at you that 6 is divisible by3, and x to the fourth is definitely divisible by xsquared, so 6x to the fourth is definitely divisibleby 3x squared. of course, it's divisible byitself, so this actually is the least common multiple-- thisis the smallest number, or the smallest expression, iguess, that is divisible by

both 6x to the fourth and 3xsquared, so let's make that the common denominator. so this sum is going to beequal to 5 over 6x fourth plus-- and now, what we wantto do is write this with 6x fourth as the denominator, solet me just write it again. so plus 7 over 3x squared. so how do we make a 3x squaredinto a 6x to the fourth? well we're going to have tomultiply it times 2 to make the 3 into a 6, and then we'regoing to have to multiply it

times another x squared, sowe're going to have to multiply it by 2x squared. now we can't just multiply onlythe denominator by 2x squared-- that'll fundamentallychange the value of this expression. we can only multiply it by 1,so let's multiply it by 2x squared over 2x squared-- andwe're assuming here that is x is not equal to 0. x does not equal to 0.

and that was actually a safeassumption to make, that 0 is not a member of our domain rightfrom the get go, because that would've made either ofthese expressions undefined. if we assume x is not equal to0, we can multiply by 2x squared over 2x squared, andthen that will give us the expression 5 over 6x to thefourth, plus-- this become 7 times 2 is 14. fourteen x squared over 3 times2 is 6, x to the squared times x squared is x to thefourth, so now we have a

common denominator. the common denominator is 6x tothe fourth, and we can just add the numerators, so it's 5plus 14x squared-- or, i like to write the higher degreeterm first-- or 14x squared plus 5. and we are assuming that x doesnot equal 0, because this would make the expressionundefined. that's about as simple as we canmake it-- we can't divide. 14 is divisible by 2, and so6, but 5 isn't, so we can't

divide everything by 2, andthen, there's x squared x to the fourth, but 5 has no x termon it, so we can't divide everything by x ora power of x. we're done: it's 14x squaredplus 5 over 6x to the fourth, and x cannot be equal to zero.