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شائع الجبر >

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

الحلّ

expand

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

الحلّ

\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}}

خطوات الحلّ

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

π (a^{2} - b^{2})

وسٌع:

π a^{2} - π b^{2}

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

π (a^{4} - b^{4})

وسٌع:

π a^{4} - π b^{4}

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

π a^{2} - π b^{2}

حلل إلى عوامل:

(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}}

π a^{4} - π b^{4}

حلل إلى عوامل:

(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} )}

(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})

المضاعف المشترك الأصغر لـ:

(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})

اكتب مجددًا الكسور بحيث يكون المقام مشترك

= \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 )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \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} )}

(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})

وسٌع:

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}}

(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)

وسٌع:

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}}

π

قم باخراج العامل المشترك

= \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}}

π^{2} a^{6} - π^{2} a^{2} b^{4} - π^{2} a^{4} b^{2} + π^{2} b^{6}

حلل إلى عوامل:

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} )}

π :

إلغ العوامل المشتركة

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

π (a^{4} - b^{4}) (a^{2} - b^{2})

وسٌع:

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}}

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