What is (a-b)/(3a+3b)-(a-2b)/(6a-6b)+(a^2+2ab-6b^2)/(9a^2-9b^2) ?

The solution to (a-b)/(3a+3b)-(a-2b)/(6a-6b)+(a^2+2ab-6b^2)/(9a^2-9b^2) is (5a)/(18(a+b))

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simplify (a-b)/(3a+3b)-(a-2b)/(6a-6b)+(a^2+2ab-6b^2)/(9a^2-9b^2)

simplify

\frac{a - b}{3 a + 3 b} - \frac{a - 2 b}{6 a - 6 b} + \frac{a ^{2} + 2 ab - 6 b ^{2}}{9 a ^{2} - 9 b ^{2}}

\frac{5 a}{18 ( a + b )}

Solution steps

\frac{a - b}{3 a + 3 b} - \frac{a - 2 b}{6 a - 6 b} + \frac{a ^{2} + 2 ab - 6 b ^{2}}{9 a ^{2} - 9 b ^{2}}

Least Common Multiplier of

3 a + 3 b, 6 a - 6 b, 9 a^{2} - 9 b^{2} : (3 a + 3 b) (6 a - 6 b) (9 a^{2} - 9 b^{2})

Adjust Fractions based on the LCM

= \frac{( a - b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} )}{( 3 a + 3 b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} )} - \frac{( a - 2 b ) ( 3 a + 3 b ) ( 9 a ^{2} - 9 b ^{2} )}{( 3 a + 3 b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} )} + \frac{( a ^{2} + 2 ab - 6 b ^{2} ) ( 3 a + 3 b ) ( 6 a - 6 b )}{( 3 a + 3 b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} )}

Apply the fraction rule:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{( a - b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} ) - ( a - 2 b ) ( 3 a + 3 b ) ( 9 a ^{2} - 9 b ^{2} ) + ( a ^{2} + 2 ab - 6 b ^{2} ) ( 3 a + 3 b ) ( 6 a - 6 b )}{( 3 a + 3 b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} )}

Simplify

(a - b) (6 a - 6 b) (9 a^{2} - 9 b^{2}) - (a - 2 b) (3 a + 3 b) (9 a^{2} - 9 b^{2}) + (a^{2} + 2 ab - 6 b^{2}) (3 a + 3 b) (6 a - 6 b) : 45 a^{4} - 45 a^{2} b^{2} - 45 a^{3} b + 45 a b^{3}

= \frac{45 a ^{4} - 45 a ^{2} b ^{2} - 45 a ^{3} b + 45 a b ^{3}}{( 3 a + 3 b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} )}

Cancel

\frac{45 a ^{4} - 45 a ^{2} b ^{2} - 45 a ^{3} b + 45 a b ^{3}}{( 3 a + 3 b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} )} : \frac{5 a}{18 ( a + b )}

= \frac{5 a}{18 ( a + b )}

(3 x + 2 y) (3 x + 2 y)

4 \cdot (- 9)

\frac{2}{2} + \frac{2}{3} + \frac{2}{4} + \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{2}{9}

\frac{cot ( x )}{sin ( x )}

(x^{2} + y^{2} + 1)^{2}

What is (a-b)/(3a+3b)-(a-2b)/(6a-6b)+(a^2+2ab-6b^2)/(9a^2-9b^2) ?
The solution to (a-b)/(3a+3b)-(a-2b)/(6a-6b)+(a^2+2ab-6b^2)/(9a^2-9b^2) is (5a)/(18(a+b))