# Simplifying this fraction

• Jun 9th 2006, 12:21 AM
sstr
Simplifying this fraction
My apologies.

64a^3 + b^3
___________
16 a^2 - b^2

Divided by

16a^2b^2 - 4ab^3 + b^4
_______________________
4a^2 - ab + 12a - 3b

Sorry I don't know how to use the math function yet.
• Jun 9th 2006, 02:17 AM
Soroban
Hello, sstr!

A fascinating problem: four types of factoring are required.

Quote:

$\displaystyle \frac{64a^3 + b^3}{16a^2 - b^2} \div \frac{16a^2b^2 - 4ab^3 + b^4}{4a^2 - ab + 12a - 3b}$

Sum of Cubes: .$\displaystyle 64a^3 + b^3 \;= \;(4a + b)(16a^2 - 4ab + b^2)$

Difference of Squares: .$\displaystyle 16a^2 - b^2 \;= \;(4a - b)(4a + b)$

Common factors: .$\displaystyle 16a^2b^2 - 4ab^3 + b^4 \;= \;b^2(16a^2 - 4ab + b^2)$

Grouping: . $\displaystyle 4a^2 - ab + 12a - 3b \;= \;a(4a - b) + 3(4a - b)\;=$ $\displaystyle \;(4a-b)(a+3)$

The problem becomes: .$\displaystyle \frac{(4a + b)(16a^2 - 4ab + b^2)}{(4a - b)(4a + b)}\; \times$ $\displaystyle \frac{(4a - b)(a + 3)}{b^2(16a^2 - 4ab + b^2)}$

. . . which reduces to: .$\displaystyle \frac{a + 3}{b^2}$