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expand 1/(pi(a^2-b^2))+(4b*c)/(pi(a^4-b^4))

Solution

expand

\frac{1}{π ( a ^{2} - b ^{2} )} + \frac{4 b \cdot c}{π ( a ^{4} - b ^{4} )}

\frac{a ^{4} - b ^{4} + 4 a ^{2} b c - 4 b ^{3} c}{π a ^{6} - π a ^{4} b ^{2} - π a ^{2} b ^{4} + π b ^{6}}

Solution steps

\frac{1}{π ( a ^{2} - b ^{2} )} + \frac{4 b c}{π ( a ^{4} - b ^{4} )}

Expand

π (a^{2} - b^{2}) : π a^{2} - π b^{2}

= \frac{1}{π a ^{2} - π b ^{2}} + \frac{4 b c}{π ( a ^{4} - b ^{4} )}

Expand

π (a^{4} - b^{4}) : π a^{4} - π b^{4}

= \frac{1}{π a ^{2} - π b ^{2}} + \frac{4 b c}{π a ^{4} - π b ^{4}}

Factor

π a^{2} - π b^{2} : (π a + π b) (π a - π b)

= \frac{1}{( π a + π b ) ( π a - π b )} + \frac{4 b c}{π a ^{4} - π b ^{4}}

Factor

π a^{4} - π b^{4} : (4 π a + 4 π b) (4 π a - 4 π b) (π a^{2} + π b^{2})

= \frac{1}{( π a + π b ) ( π a - π b )} + \frac{4 b c}{( 4 π a + 4 π b ) ( 4 π a - 4 π b ) ( π a ^{2} + π b ^{2} )}

Least Common Multiplier of

(π a + π b) (π a - π b), (4 π a + 4 π b) (4 π a - 4 π b) (π a^{2} + π b^{2}) : (π a + π b) (π a - π b) (4 π a + 4 π b) (4 π a - 4 π b) (π a^{2} + π b^{2})

Adjust Fractions based on the LCM

= \frac{( 4 π a + 4 π b ) ( 4 π a - 4 π b ) ( π a ^{2} + π b ^{2} )}{( π a + π b ) ( π a - π b ) ( 4 π a + 4 π b ) ( 4 π a - 4 π b ) ( π a ^{2} + π b ^{2} )} + \frac{4 b c ( π a + π b ) ( π a - π b )}{( 4 π a + 4 π b ) ( 4 π a - 4 π b ) ( π a ^{2} + π b ^{2} ) ( π a + π b ) ( π a - π b )}

Since the denominators are equal, combine the fractions:

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

= \frac{( 4 π a + 4 π b ) ( 4 π a - 4 π b ) ( π a ^{2} + π b ^{2} ) + 4 b c ( π a + π b ) ( π a - π b )}{( π a + π b ) ( π a - π b ) ( 4 π a + 4 π b ) ( 4 π a - 4 π b ) ( π a ^{2} + π b ^{2} )}

Expand

(π a + π b) (π a - π b) (4 π a + 4 π b) (4 π a - 4 π b) (π a^{2} + π b^{2}) : π^{2} a^{6} - π^{2} a^{2} b^{4} - π^{2} a^{4} b^{2} + π^{2} b^{6}

= \frac{( 4 π a + 4 π b ) ( 4 π a - 4 π b ) ( π a ^{2} + π b ^{2} ) + 4 b c ( π a + π b ) ( π a - π b )}{π ^{2} a ^{6} - π ^{2} a ^{2} b ^{4} - π ^{2} a ^{4} b ^{2} + π ^{2} b ^{6}}

Expand

(4 π a + 4 π b) (4 π a - 4 π b) (π a^{2} + π b^{2}) + 4 b c (π a + π b) (π a - π b) : π a^{4} - π b^{4} + 4 π a^{2} b c - 4 π b^{3} c

= \frac{π a ^{4} - π b ^{4} + 4 π a ^{2} b c - 4 π b ^{3} c}{π ^{2} a ^{6} - π ^{2} a ^{2} b ^{4} - π ^{2} a ^{4} b ^{2} + π ^{2} b ^{6}}

Factor out common term

π

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

Factor

π^{2} a^{6} - π^{2} a^{2} b^{4} - π^{2} a^{4} b^{2} + π^{2} b^{6} : π^{2} (a^{4} - b^{4}) (a^{2} - b^{2})

= \frac{π ( a ^{4} - b ^{4} + 4 a ^{2} b c - 4 b ^{3} c )}{π ^{2} ( a ^{4} - b ^{4} ) ( a ^{2} - b ^{2} )}

Cancel the common factor:

π

= \frac{a ^{4} - b ^{4} + 4 a ^{2} b c - 4 b ^{3} c}{π ( a ^{4} - b ^{4} ) ( a ^{2} - b ^{2} )}

Expand

π (a^{4} - b^{4}) (a^{2} - b^{2}) : π a^{6} - π a^{4} b^{2} - π a^{2} b^{4} + π b^{6}

= \frac{a ^{4} - b ^{4} + 4 a ^{2} b c - 4 b ^{3} c}{π a ^{6} - π a ^{4} b ^{2} - π a ^{2} b ^{4} + π b ^{6}}

e x p an d \frac{( 2 - x ) ^{3}}{6}

e x p an d a (- 5, 2) b (4, - 3)

s im pl i f y (x - 3 y) (x + 3 y)

e x p an d α (2, 1, - 2)

s im pl i f y \frac{k ( s + 1.5 )}{s ( s + 1 ) ( s + 10 ) + 1.5 k}