If a + b = 7 and a^3 + b^3 = 45, what is the value of the sum 1/a + 1/b? Express your answer as a common fraction.

 Apr 4, 2021

Using the sum of cubes factorization on $a^3+b^3=45$ yields $(a+b)(a^2-ab+b^2)=45$, which gives $7(a^2-ab+b^2)=45$, which gives $a^2-ab+b^2=\frac{45}{7}$


Squaring $a+b=7$ gives $a^2+2ab+b^2=49$. We now have a system of equations! From here, there are two ways to proceed. One is to solve the system of equations, find the values of a and b, and plug them in. The other way, which is MUCH faster, will be shown here.


Subtracting the first equation from the second yields $2ab-(-ab)=\frac{298}{7}$, which gives $ab=\frac{298}{21}$


Now, we make use of the fact that $\frac{1}{a}+\frac{1}{b}=\frac{a+b}{ab}$(you can check this yourself by splitting up the fractions), and we plug in! We now have $\frac{a+b}{ab}=\frac{7}{\frac{298}{21}}=\boxed{\frac{147}{298}}$

 Apr 4, 2021

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