Simplify each complex fraction. [ 1/(a^3+b^3) ] / [ 1/(a^2 + 2ab ...

Question

Simplify each complex fraction. [ 1/(a^3+b^3) ] / [ 1/(a^2 + 2ab + b^2) ]

Accepted Answer

Hey everyone here we are asked to simplify the following expression where we have one divided by c cubed minus D cubed all divided by one divided by c squared minus two C. D plus D squared. So now before we begin simplifying we first need to notice that our given expression here is a complex fraction because we have a fraction in both the numerator and the denominator. So to begin simplifying we first need to recall that when we divide by fractions we need to multiply by the reciprocal so we can use this fact to rewrite our X expression as one divided by C cubed minus D cubed times the reciprocal of the fraction in the denominator which is C squared minus two C. D plus D squared all divided by one. And now we can simplify this expression by noticing that we have a one in the first numerous and a one in the second denominator. So our expression now becomes C squared minus two C. D plus d squared, all divided by c cubed minus d cubed. And now we can begin our simplification process by first factoring both the numerator and and the denominator. So beginning with our numerator. Well this expression of C squared minus two C. D plus D squared just simply becomes the squared expression of the quantity of C minus D squared. And now working on factoring our denominator, we need to notice that we have the difference of two cubes here. So we need to recall the difference of cubes formula, which tells us that X cubed minus Y cubed is equal to the quantity of x minus Y times the quantity of X squared plus X, Y plus Y squared. And now comparing this formula with our denominator we see that the value for X is going to be C and the value for Y is going to be D. And now using the right hand side of our formula, we see that our denominator becomes the quantity of c minus D times the quantity of C squared plus C. D plus Y. Sorry plus D squared. So now we can see that we can simplify our expression further since we have this quantity of C minus D. In the denominator and we have the quantity of C minus D squared in the numerator. So the c minus D. In the denominator cancels out. But the c minus D. In the numerator just has its power reduced to one. So now we can really right our simplified expression as the quantity of C minus D divided by C squared plus C. D plus D squared. And now there aren't any other simplifications that we can make here. So we have found our final simplified form which leaves us with answer choice C. Thanks for watching. I hope you found this video helpful and I'll see you again next time

Simplify each complex fraction. [ 1/(a^3+b^3) ] / [ 1/(a^2 + 2ab + b^2) ]

Key Concepts

Complex Fractions

Factoring Polynomials

Division of Fractions

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