1–2. Дараҷаи нишондиҳандааш натуралӣ ва бутун - Алгебра — синфи 10
Дараҷаи нишондиҳандааш натуралӣ
Масалан, 2 3 = 2 ⋅ 2 ⋅ 2 = 8 , 2^3 = 2 \cdot 2 \cdot 2 = 8, 2 3 = 2 ⋅ 2 ⋅ 2 = 8 , ( − 1 3 ) 4 = 1 81 . \left(-\dfrac{1}{3}\right)^4 = \dfrac{1}{81}. ( −
навбатӣ 3. Решаи дараҷаи n-ум ва хосиятҳои он 1 соат
3
1
)
4
=
81 1 .
Эзоҳ. Ифодаҳои ( − a ) n (-a)^n ( − a ) n − a n -a^n − a n ҳамаи − a -a − a a , a, a ,
Масалан, 2 2 ⋅ 2 3 = 2 5 = 32 , 2^2 \cdot 2^3 = 2^5 = 32, 2 2 ⋅ 2 3 = 2 5 = 32 , 3 8 : 3 10 = 3 − 2 , 3^8 : 3^{10} = 3^{-2}, 3 8 : 3 10 = 3 − 2 , ( x + y ) 8 : ( x + y ) 5 = ( x + y ) 3 . (x+y)^8 : (x+y)^5 = (x+y)^3. ( x + y ) 8 : ( x + y ) 5 = ( x + y ) a m ⋅ a n ⋅ a k = a m + n + k . a^m \cdot a^n \cdot a^k = a^{m+n+k}. a m ⋅ a n ⋅ a k = a m + n + k
Баъзан баробариро аз рост ба чап истифода бурдан қулай аст. Масалан, M = 8 3 ⋅ 25 3 ⋅ 2 3 M = 8^3 \cdot 25^3 \cdot 2^3 M = 8 3 ⋅ 2 5 3 ⋅ 2 3 3 ∘ 3^\circ 3 ∘ M = ( 8 ⋅ 25 ⋅ 2 ) 3 = 400 3 = 64 000 000. M = (8 \cdot 25 \cdot 2)^3 = 400^3 = 64\,000\,000. M = ( 8 ⋅ 25 ⋅ 2 ) 3 = 40 0 3 = 64 000 000.
Масалан, ( 2 3 ) 4 = 16 81 , \left(\dfrac{2}{3}\right)^4 = \dfrac{16}{81}, ( 3 2 ) 4 = 81 16 , ( − 5 4 ) 3 = − 125 64 , \left(-\dfrac{5}{4}\right)^3 = -\dfrac{125}{64}, ( − 4 5 ) 3 = − 64 125 , ( 2 a 3 b ) 3 = 8 a 3 27 b 3 . \left(\dfrac{2a}{3b}\right)^3 = \dfrac{8a^3}{27b^3}. ( 3 b 2 a ) 3 = 27 b
Масалан, ( 2 3 ) 2 = 2 6 = 64 , (2^3)^2 = 2^6 = 64, ( 2 3 ) 2 = 2 6 = 64 , ( − 1 2 a 4 b 2 ) 3 = − 1 8 a 12 b 6 . \left(-\dfrac{1}{2}a^4b^2\right)^3 = -\dfrac{1}{8}a^{12}b^6. ( − 2 1 a 4 b 2 ) 3 = − 8 1 a 12 b 6 .
Квадрати бисёраъзогӣ Квадрати суммаи якчанд аъзо ба суммаи квадратҳои ҳамаи аъзоҳо плюс ҳосили зарби дучандаи ҳар як аъзо бар ҳамаи аъзоҳои пасоянд баробар аст:
( a + b + c + d ) 2 = a 2 + b 2 + c 2 + d 2 + 2 a b + 2 a c + 2 a d + 2 b c + 2 b d + 2 c d . (a + b + c + d)^2 = a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd. ( a + b + c + d ) 2 = a 2 + b 2 + c 2 + d 2 + 2 ab + 2 a c + 2 a d + 2 b c + 2 b d + 2 c d .
Мисоли 1. ( 3 x 2 + 2 y 2 + x y ) 2 (3x^2 + 2y^2 + xy)^2 ( 3 x 2 + 2 y 2 + x y ) 2 ( 3 x 2 + 2 y 2 + x y ) 2 = 9 x 4 + 4 y 4 + x 2 y 2 + 2 ⋅ 3 x 2 ⋅ 2 y 2 + 2 ⋅ 3 x 2 ⋅ x y + 2 ⋅ 2 y 2 ⋅ x y = 9 x 4 + 4 y 4 + 13 x 2 y 2 + 6 x 3 y + 4 x y 3 . \begin{aligned} (3x^2 + 2y^2 + xy)^2 &= 9x^4 + 4y^4 + x^2y^2 + 2 \cdot 3x^2 \cdot 2y^2 + 2 \cdot 3x^2 \cdot xy + 2 \cdot 2y^2 \cdot xy \\ &= 9x^4 + 4y^4 + 13x^2y^2 + 6x^3y + 4xy^3. \end{aligned} ( 3 x 2 + 2 y 2 + x y ) 2 = 9 x 4
Мисоли 2. ( a − 2 b + 3 c − 4 d ) 2 (a - 2b + 3c - 4d)^2 ( a − 2 b + 3 c − 4 d ) 2 (a , a, a , − 2 b , -2b, − 2 b , 3 c , 3c, 3 c , − 4 d -4d − 4 d a 2 + 4 b 2 + 9 c 2 + 16 d 2 + 2 a ( − 2 b ) + 2 a ⋅ 3 c + 2 a ( − 4 d ) + 2 ( − 2 b ) ⋅ 3 c + 2 ( − 2 b ) ( − 4 d ) + 2 ⋅ 3 c ( − 4 d ) = a 2 + 4 b 2 + 9 c 2 + 16 d 2 − 4 a b + 6 a c − 8 a d − 12 b c + 16 b d − 24 c d . \begin{aligned} &a^2 + 4b^2 + 9c^2 + 16d^2 + 2a(-2b) + 2a \cdot 3c + 2a(-4d) + 2(-2b) \cdot 3c + 2(-2b)(-4d) + 2 \cdot 3c(-4d) \\ &= a^2 + 4b^2 + 9c^2 + 16d^2 - 4ab + 6ac - 8ad - 12bc + 16bd - 24cd. \end{aligned}
Дараҷаи нишондиҳандааш нул ва бутуни манфӣ Масалан, 10 − 2 = 0,01 , 10^{-2} = 0{,}01, 1 0 − 2 = 0 , 01 , ( 2 3 ) − 2 = 1 ( 2 3 ) 2 = 9 4 = 2,25. \left(\dfrac{2}{3}\right)^{-2} = \dfrac{1}{\left(\frac{2}{3}\right)^2} = \dfrac{9}{4} = 2{,}25. ( 3 2 ) − 2 = ( 3 2 ) 2 1 = 4 9 = 2 , 25. 1 1 1 1 3 = 3 − 1 , \dfrac{1}{3} = 3^{-1}, 3 1 = 3 − 1 , 1 8 = 1 2 3 = 2 − 3 . \dfrac{1}{8} = \dfrac{1}{2^3} = 2^{-3}. 8 1 = 2 3 1 = 2
Ин таърифҳо тавре интихоб шудаанд, ки қоидаҳои 2 ∘ 2^\circ 2 ∘ 5 ∘ 5^\circ 5 ∘ ҳамаи нишондиҳандаҳои бутун дуруст монанд. Ҳангоми a ≠ 0 a \ne 0 a = 0 m , m, m , n n n a m ⋅ a n = a m + n , a m : a n = a m − n , ( a m ) n = a m n . a^m \cdot a^n = a^{m+n}, \qquad a^m : a^n = a^{m-n}, \qquad (a^m)^n = a^{mn}. a m ⋅ a n = a m + n , a n , n, n , m > 0 m > 0 m > 0
Тақсим бо зарб иваз мешавад: a m : a n = a m ⋅ 1 a n = a m ⋅ a − n = a m − n . a^m : a^n = a^m \cdot \dfrac{1}{a^n} = a^m \cdot a^{-n} = a^{m-n}. a m : a n = a m ⋅ a n 1 = a m ⋅ a − n = a m − n . ( a − n ) m = ( 1 a n ) m = 1 a n m = a − n m , (a^{-n})^m = \left(\dfrac{1}{a^n}\right)^m = \dfrac{1}{a^{nm}} = a^{-nm}, ( a − n ) m = ( a n ( a n ) − m = 1 ( a n ) m = a − n m , (a^n)^{-m} = \dfrac{1}{(a^n)^m} = a^{-nm}, ( a n ) − m = ( a n ) ( a − n ) − m = 1 ( a − n ) m = 1 a − n m = a n m . (a^{-n})^{-m} = \dfrac{1}{(a^{-n})^m} = \dfrac{1}{a^{-nm}} = a^{nm}. ( a − n ) − m = ( a
Мисоли 3. Ҳисоб мекунем:
2 − 3 ⋅ 16 2 ( 4 − 1 ) 3 = 2 − 3 ⋅ 2 8 4 − 3 = 2 5 2 − 6 = 2 5 − ( − 6 ) = 2 11 = 2048. \frac{2^{-3} \cdot 16^2}{(4^{-1})^3} = \frac{2^{-3} \cdot 2^8}{4^{-3}} = \frac{2^5}{2^{-6}} = 2^{5 - (-6)} = 2^{11} = 2048. ( 4 − 1 ) 3 2 − 3 ⋅ 1 6 2 = 4 − 3 2 − 3 ⋅ 2 8 = 2 − 6 2 5 = 2 5 − ( − 6 ) = 2 11 = 2048.
Мисоли 4. Ифодаро содда мекунем (ҳангоми a , a, a , b ≠ 0 , b \ne 0, b = 0 , a + b ≠ 0 a + b \ne 0 a + b = 0 ( a − 1 + b − 1 ) ( a + b ) − 1 = a + b a b ⋅ 1 a + b = 1 a b = a − 1 b − 1 . (a^{-1} + b^{-1})(a + b)^{-1} = \frac{a + b}{ab} \cdot \frac{1}{a + b} = \frac{1}{ab} = a^{-1}b^{-1}. ( a − 1 + b − 1 ) ( a + b ) − 1 = ab a +
Кор дар синф
Машқҳо
Дар синф 1. Масоҳати квадратеро ҳисоб кунед, ки тарафҳояш ба 5 5 5 10 10 10 100 100 100
2. Формулаҳои ( a + b ) 2 = a 2 + 2 a b + b 2 , (a + b)^2 = a^2 + 2ab + b^2, ( a + b ) 2 = a 2 + 2 ab + b 2 , ( a − b ) 2 = a 2 − 2 a b + b 2 (a - b)^2 = a^2 - 2ab + b^2 ( a − b ) 2 = a 2 − 2 ab + b 2 a 2 − b 2 = ( a − b ) ( a + b ) a^2 - b^2 = (a - b)(a + b) a 2 − b 2 = ( a − b ) ( a + b ) 31 2 ; 51 2 ; 42 2 ; 47 2 − 23 2 ; 84 2 − 46 2 . 31^2;\quad 51^2;\quad 42^2;\quad 47^2 - 23^2;\quad 84^2 - 46^2. 3 1 2 ; 5 1 2 ; 4 2 2 ; 4 7 2 −
3. а) Ифодаҳоро ҳисоб кунед:
3 − 2 ; 3^{-2}; 3 − 2 ;
( − 1 2 ) − 3 ; \left(-\dfrac{1}{2}\right)^{-3}; ( − 2 1 ) − 3 ;
( 2 5 ) − 2 ; \left(\dfrac{2}{5}\right)^{-2}; ( 5 2 ) − 2 ;
( 2 5 ) − 2 ⋅ 4 8 ; (2^5)^{-2} \cdot 4^8; ( 2 5 ) − 2 ⋅ 4 8 ;
( 3 2 ) − 4 ; (3^2)^{-4}; ( 3 2 ) − 4 ;
[ ( 1 5 ) − 2 ] 3 ; \left[\left(\dfrac{1}{5}\right)^{-2}\right]^3; [ ( 5 1 ) − 2 ] 3 ;
[ a − ( 1 − a ) − 1 ] ( a − 2 ) + a 0 1 a − 2 − a + 1 ; \left[a - (1 - a)^{-1}\right] \dfrac{(a - 2) + a^0}{\dfrac{1}{a^{-2}} - a + 1}; [ a − ( 1 − a ) − 1 ] a
( x − 2 + a − 3 ) ( x − 2 − a − 3 ) ; (x^{-2} + a^{-3})(x^{-2} - a^{-3}); ( x − 2 + a − 3 ) ( x − 2 − a
a − 2 b − 1 + a − 1 b − 2 a − 2 b − 2 + a 3 ( a 2 − 2 a b + b 2 ) − 2 . \dfrac{a^{-2}b^{-1} + a^{-1}b^{-2}}{a^{-2}b^{-2}} + a^3(a^2 - 2ab + b^2)^{-2}. a − 2 b − 2 a
б) Ифодаҳоро содда намоед:
4 a − 3 b 2 c − 1 ⋅ 0,25 a b − 5 c − 2 ; 4a^{-3}b^2c^{-1} \cdot 0{,}25ab^{-5}c^{-2}; 4 a − 3 b 2 c − 1 ⋅ 0 , 25 a b − 5 c − 2 ;
6 a 5 x 7 z − 8 4 − 1 a − 8 x − 4 z ; \dfrac{6a^5x^7z^{-8}}{4^{-1}a^{-8}x^{-4}z}; 4 − 1 a − 8 x − 4 z 6 a
( − 3 ⋅ 2 − 1 a m − n x ) − 2 ; \left(-3 \cdot 2^{-1}am^{-n}x\right)^{-2}; ( − 3 ⋅ 2 − 1 a m − n x ) − 2 ;
( 4 a − 2 − b − 4 ) : ( 2 b 2 − a ) ; (4a^{-2} - b^{-4}) : (2b^2 - a); ( 4 a − 2 − b − 4 ) : ( 2 b 2 − a ) ;
( x − 2 + 1 ) − 2 ; (x^{-2} + 1)^{-2}; ( x − 2 + 1 ) − 2 ;
( 8 ⋅ 0,25 2 − x + 6 ⋅ 2 2 x − 0,5 − 2 x + 1 ) : 2 2 x − 3 . \left(8 \cdot 0{,}25^{2-x} + 6 \cdot 2^{2x} - 0{,}5^{-2x+1}\right) : 2^{2x-3}. ( 8 ⋅ 0 , 2 5 2 − x + 6 ⋅ 2 2 x − 0 , 5
Дар хона
3 x n − 1 ⋅ 2 x n + 1 ; 3x^{n-1} \cdot 2x^{n+1}; 3 x n − 1 ⋅ 2 x n + 1 ;
( 2 x n − 1 ) n + 1 ; \left(2x^{n-1}\right)^{n+1}; ( 2 x n − 1 ) n + 1 ;
( a 2 x − 1 − 2 a x + 2 + 3 a x + 9 ) : 0,1 a x − 2 ; \left(a^{2x-1} - 2a^{x+2} + 3a^{x+9}\right) : 0{,}1a^{x-2}; ( a 2 x − 1 − 2 a x + 2 + 3 a x + 9
35 ⋅ ( 27 4 + 2 ⋅ 9 6 ) ( 81 3 − 12 ⋅ 3 11 ) ⋅ 15 . \dfrac{35 \cdot (27^4 + 2 \cdot 9^6)}{(81^3 - 12 \cdot 3^{11}) \cdot 15}. ( 8 1 3 − 12 ⋅ 3 11 ) ⋅ 15 35 ⋅ (
5. Агар n n n 2 n 2n 2 n 2 n + 1 2n + 1 2 n + 1
6. Айниятҳоро исбот кунед:
( m + n ) 2 2 + ( m − n ) 2 2 = m 2 + n 2 ; \dfrac{(m + n)^2}{2} + \dfrac{(m - n)^2}{2} = m^2 + n^2; 2 ( m + n ) 2 + 2 ( m − n ) 2 = m 2 + n 2 ;
( m + n 2 ) 2 − ( m − n 2 ) 2 = m n ; \left(\dfrac{m + n}{2}\right)^2 - \left(\dfrac{m - n}{2}\right)^2 = mn; ( 2 m + n ) 2 − (
( a 2 − 1 a 2 + 1 ) 2 + ( 2 a a 2 + 1 ) 2 = 1. \left(\dfrac{a^2 - 1}{a^2 + 1}\right)^2 + \left(\dfrac{2a}{a^2 + 1}\right)^2 = 1. ( a 2 + 1 a 2 − 1
7. Ба зарбкунандаҳо ҷудо кунед:
m 4 + m 3 + m + 1 ; m^4 + m^3 + m + 1; m 4 + m 3 + m + 1 ;
n 4 + n 3 − n − 1 ; n^4 + n^3 - n - 1; n 4 + n 3 − n − 1 ;
( x + y ) 3 − ( x − y ) 3 ; (x + y)^3 - (x - y)^3; ( x + y ) 3 − ( x − y ) 3 ;
( x + y ) 4 − ( x − y ) 4 ; (x + y)^4 - (x - y)^4; ( x + y ) 4 − ( x − y ) 4 ;
a 2 − a − 12 ; a^2 - a - 12; a 2 − a − 12 ;
m 2 + 3 m − 10 ; m^2 + 3m - 10; m 2 + 3 m − 10 ;
2 x 2 + 10 x + 12. 2x^2 + 10x + 12. 2 x 2 + 10 x + 12.
3 1 2 − 7 2 3 + 1 2 − 2 3 ; 3\dfrac{1}{2} - 7\dfrac{2}{3} + \dfrac{1}{2} - \dfrac{2}{3}; 3 2 1 − 7 3 2 + 2 1 − 3 2 ;
7 1 5 − 2 3 ( 7 1 5 − 4 2 3 ) ; 7\dfrac{1}{5} - \dfrac{2}{3}\left(7\dfrac{1}{5} - 4\dfrac{2}{3}\right); 7 5 1 − 3 2 ( 7
3 5 ⋅ ( − 1 2 ) ⋅ ( − 1 4 ) . \dfrac{3}{5} \cdot \left(-\dfrac{1}{2}\right) \cdot \left(-\dfrac{1}{4}\right). 5 3 ⋅ ( − 2 1 ) ⋅
Вазифаи хонагӣ
Машқҳои 4–8.
Вазифаи хонагии онлайн: «Алгебра 10 · Вазифаи хонагӣ: 1–2» дар охири ин дарс.
Хулоса
a n a^n a n n n n a a a a 0 = 1 a^0 = 1 a 0 = 1 (a ≠ 0 a \ne 0 a = 0 0 0 0^0 0 0 a − n = 1 a n a^{-n} = \dfrac{1}{a^n} a − n = a n 1 Барои нишондиҳандаҳои бутун ҳамон се қоида дуруст аст: a m a n = a m + n , a^m a^n = a^{m+n}, a m a n = a m + n , a m : a n = a m − n , a^m : a^n = a^{m-n}, a
Квадрати бисёраъзогӣ = суммаи квадратҳо + ҳосили зарби дучандаи ҳар ҷуфти аъзоҳо.
Дар ин дарс
Ман медонам таърифи дараҷаи нишондиҳандааш натуралӣ, нул ва бутуни манфӣ ва қоидаҳои амалҳо бо дараҷаҳо.Ман метавонам ифодаҳоро бо дараҷаҳои нишондиҳандаашон бутун ҳисоб кунам ва сода намоям.Ман метавонам квадрати бисёраъзогиро кушоям ва ба зарбкунандаҳо ҷудо кунам. Такрор (2 дақиқа)
( − 2 ) 3 (-2)^3 ( − 2 ) 3 − 2 3 -2^3 − 2 3 5 4 : 5 2 5^4 : 5^2 5 4 : 5 2 ( a + b ) 2 (a + b)^2 ( a + b ) 2 Ҷавобҳо
( − 2 ) 3 = − 8 (-2)^3 = -8 ( − 2 ) 3 = − 8 − 2 3 = − 8 -2^3 = -8 − 2 3 = − 8 2 3 2^3 2 Вазъият барои андешаронӣ Масофаи Замин то Офтоб тақрибан 150 000 000 150\,000\,000 150 000 000 0,000000001 0{,}000000001 0 , 000000001 10 8 10^8 1 0 8 10 − 9 10^{-9} 1 0 − 9 10 − 9 10^{-9} 1 0 − 9 10 8 10^8 1 0 8
Таъриф Дараҷаи адади a a a n n n n n n a a a a n = a ⋅ a ⋅ … ⋅ a ⏟ n маротиба . a^n = \underbrace{a \cdot a \cdot \ldots \cdot a}_{n\text{ маротиба}}. a n = n маротиба a ⋅ a ⋅ … ⋅ a . a n a^n a n a a a асос , n n n нишондиҳанда ном дорад. Дараҷаи якум: a 1 = a . a^1 = a. a 1 = a .
Хосият 1° Дараҷаи ҷуфти адади мусбат ё манфӣ мусбат аст, дараҷаи тоқи адади манфӣ манфӣ:
( ± a ) 2 n = a 2 n , ( ± a ) 2 n + 1 = ± a 2 n + 1 ( a > 0 ) . (\pm a)^{2n} = a^{2n}, \qquad (\pm a)^{2n+1} = \pm a^{2n+1} \quad (a > 0). ( ± a ) 2 n = a 2 n , ( ± a )
Хосият 2° Ҳангоми зарби дараҷаҳои асосашон якхела нишондиҳандаҳо ҷамъ, ҳангоми тақсим — тарҳ карда мешаванд:
a m ⋅ a n = a m + n , a m : a n = a m − n . a^m \cdot a^n = a^{m+n}, \qquad a^m : a^n = a^{m-n}. a m ⋅ a n = a m + n ,
Хосият 3° Ҳангоми ба дараҷа бардоштани ҳосили зарб ҳар як зарбшаванда ба ҳамон дараҷа бардошта мешавад:
( a b c ) n = a n ⋅ b n ⋅ c n . (abc)^n = a^n \cdot b^n \cdot c^n. ( ab c ) n = a n ⋅ b n ⋅ c
Хосият 4° Барои ба дараҷа бардоштани каср ҳам суратро, ҳам махраҷро ба ҳамон дараҷа бардошта, натиҷаи якумро ба дуюм тақсим мекунанд:
( a b ) n = a n b n ( b ≠ 0 ) . \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \ne 0). ( b a ) n =
Хосият 5° Ҳангоми ба дараҷа бардоштани дараҷа нишондиҳандаҳо зарб карда мешаванд:
( a n ) m = a n m . (a^n)^m = a^{nm}. ( a n ) m = a nm .
Таъриф 1 Ҳар як адади a ≠ 0 a \ne 0 a = 0 a 0 = 1. a^0 = 1. a 0 = 1. 0 0 0^0 0 0
Таъриф 2 Дараҷаи адади a ≠ 0 a \ne 0 a = 0 − n -n − n 1 1 1 a n a^n a n
Зарби дараҷаҳо
Як нишондиҳанда нул аст: a 0 ⋅ a − n = 1 ⋅ 1 a n = a − n = a 0 + ( − n ) . a^0 \cdot a^{-n} = 1 \cdot \dfrac{1}{a^n} = a^{-n} = a^{0 + (-n)}. a 0 ⋅ a − n = 1 ⋅ a
Калимаҳои калидӣ дараҷа асос нишондиҳанда дараҷаи нишондиҳандааш бутун квадрати бисёраъзогӣ
Ман дар ин дарс чӣ омӯхтам? Ман метавонам:
✓ / ✗ таърифи a 0 a^0 a 0 a − n a^{-n} a − n
✓ / ✗ нишондиҳандаҳои дараҷаҳоро дар зарб, тақсим ва дараҷа ба дараҷа дуруст ҳисоб кунам;
✓ / ✗ фарқи ( − a ) n (-a)^n
( − 2 ) 4 = − 2 4 . (-2)^4 = -2^4. ( − 2 ) 4 = − 2 4 .
( a − 2 ) 3 (a^{-2})^3 ( a − 2 ) 3 (a ≠ 0 a \ne 0 a = 0
Як ҷавобро интихоб кунед A a − 5 a^{-5} a
B C D Кадоме аз ифодаҳои зерин маъно надорад?
Як ҷавобро интихоб кунед A 0 5 0^5 0 5
B 2 0 2^0 2 0
C 0 0 0^0 0 0
D ( − 3 ) 0 (-3)^0 ( − 3 )
Ифодаро бо қимати он мувофиқ кунед.
Ҳар як унсури чапро бо яке аз рост мувофиқ кунед 2 − 3 2^{-3} 2 − 3
→ … −8 −1/8 1/8 1/3 9/4 4/9 ( − 1 2 ) − 3 \left(-\dfrac{1}{2}\right)^{-3} ( − 2 1 ) − 3
→ … −8 −1/8 1/8 1/3 9/4 4/9 → … −8 −1/8 1/8 1/3 9/4 4/9
3
.
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= 9 x 4 + 4 y 4 + 13 x 2 y 2 + 6 x 3 y + 4 x y 3 .
a 2 + 4 b 2 + 9 c 2 + 16 d 2 + 2 a ( − 2 b ) + 2 a ⋅ 3 c + 2 a ( − 4 d ) + 2 ( − 2 b ) ⋅ 3 c + 2 ( − 2 b ) ( − 4 d ) + 2 ⋅ 3 c ( − 4 d ) = a 2 + 4 b 2 + 9 c 2 + 16 d 2 − 4 ab + 6 a c − 8 a d − 12
− 3
.
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:
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1
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+
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m
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3
( − 2 ) 4 = 16 , (-2)^4 = 16, ( − 2 ) 4 = 16 , − 2 4 = − 16. -2^4 = -16. − 2 4 = − 16. a 2 + 2 a b + b 2 . a^2 + 2ab + b^2. a 2 + 2 ab + b 2 . 2 n + 1
=
± a 2 n + 1 ( a >
0 ) .
( − 3 ) 4 = 81 , (-3)^4 = 81, ( − 3 ) 4 = 81 , ( − 2 ) 5 = − 32. (-2)^5 = -32. ( − 2 ) 5 = − 32. a m
:
a n =
a m − n .
n
.
b n a n
(
b
=
0 ) .
a − n = 1 a n . a^{-n} = \frac{1}{a^n}. a − n = a n 1 .
n
1
=
a − n =
a 0 + ( − n ) .
Ҳарду манфӣ: a − n ⋅ a − m = 1 a n ⋅ 1 a m = 1 a n + m = a − ( n + m ) . a^{-n} \cdot a^{-m} = \dfrac{1}{a^n} \cdot \dfrac{1}{a^m} = \dfrac{1}{a^{n+m}} = a^{-(n+m)}. a − n ⋅ a − m = a n 1 ⋅ a m 1 = a n + m 1 = a − ( n + m ) . Яке мусбат, дигаре манфӣ: a m ⋅ a − n = a m ⋅ 1 a n = a m a n = a m − n a^m \cdot a^{-n} = a^m \cdot \dfrac{1}{a^n} = \dfrac{a^m}{a^n} = a^{m-n} a m ⋅ a − n = a m ⋅ a n 1 = a n a m = a m − n m > n m > n m > n 2 ∘ 2^\circ 2 ∘ m < n m < n m < n 1 a n − m = a m − n \dfrac{1}{a^{n-m}} = a^{m-n} a n − m 1 = a m − n (
−
a
) n
✓ / ✗ квадрати суммаи се аъзоро кушоям. Дар кадом қисм хато мекунам? Бо кадом қадам ба худ кумак мекунам?
− 5
0
5 0 ⋅ 3 − 1
( 2 3 ) − 2 \left(\dfrac{2}{3}\right)^{-2} ( 3 2 ) − 2
b
c
+
16
b
d
−
24
c
d
.