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14.1.6. Ïðîïîðö³¿<br />
a c<br />
= ⇔ ad = bc .<br />
b d<br />
(a, d — êðàéí³ ÷ëåíè ïðîïîðö³¿, b, c — ñåðåäí³ ÷ëåíè ïðîïîðö³¿)<br />
òà ¿õí³ âëàñòèâîñò³:<br />
a+ b c+ d a−b c− d a+ b c+ d a−b c−d<br />
= ; = ; = ; = ;<br />
a c a c b d b d<br />
a a<br />
ab = a b ; ( b );<br />
b<br />
= b<br />
≠0<br />
a+ b ≤ a + b ; a−b ≥ a − b .<br />
14.2.2. Ôîðìóëè ñêîðî÷åíîãî ìíîæåííÿ:<br />
( )<br />
2 2 2<br />
a± b = a ± 2 ab+<br />
b ;<br />
a = c ; a = c ; b = d ;<br />
b =<br />
d<br />
a+ b c+ d a−b c− d a+ b c+ d a−b c−d<br />
a+ b c+ d a+ c a c a+<br />
b a b<br />
= ; = = ; = = ;<br />
a−b c− d b+ d b d c+<br />
d c d<br />
a−c a c a−b a b<br />
= = ; = = .<br />
b−d b d c−d c d<br />
Çàãàëüí³ ïîõ³äí³ ïðîïîðö³¿:<br />
ma + nb mc + nd ma + nc mb + nd<br />
= ; =<br />
ma+ nb mc+ nd ma+ nc mb+<br />
nd<br />
1 1 1 1 1 1 1 1<br />
äå m, n, m 1 , n 1 — äîâ³ëüí³ ÷èñëà.<br />
14.2. ÀËÃÅÁÐÀ<br />
;<br />
( )<br />
( )<br />
3 3 2 2 3<br />
a± b = a ± 3a b+ 3 ab ± b ;<br />
( )( )<br />
2 2<br />
a − b = a− b a+<br />
b ;<br />
( )( )<br />
3 3 2 2<br />
a ± b = a± b a m ab+<br />
b ;<br />
2 2 2 2<br />
a + b + c = a + b + c + 2ab + 2ac + 2 bc .<br />
14.2.3. Ñòåïåí³ òà ¿õí³ âëàñòèâîñò³:<br />
( )<br />
n<br />
0 1<br />
a = a14243 ⋅a⋅K⋅ a; a = 1 a ≠ 0 ; a = a<br />
;<br />
1<br />
n<br />
n<br />
m<br />
n<br />
1<br />
n n m −n<br />
a = a; a = a ; a = ( m,<br />
n∈N n<br />
);<br />
a<br />
14.2.1. Ìîäóëü ä³éñíîãî ÷èñëà ³ éîãî âëàñòèâîñò³:<br />
p p p p<br />
q<br />
pq p q p<br />
( ) = ; ( ) = ; = + q<br />
( , ∈ )<br />
ab a b a a a a a p q R ;<br />
a<br />
⎧ a, a ≥ 0,<br />
= ⎨<br />
⎩ − a, a < 0;<br />
p p p<br />
⎛a⎞<br />
a a p−q<br />
⎜ ⎟ = ; = a ( p, q∈R , b ≠0<br />
p q<br />
).<br />
⎝b⎠<br />
b a<br />
a ≥ 0; a = 0⇔ a = 0;<br />
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