Stepenovanje sa racionalnim izložiocem
Uvod
Već znamo šta znači a n a^n a n n n n 4 1 2 4^{\frac{1}{2}} 4 2 1 8 2 3 8^{\frac{2}{3}} 8 3 2
Sadržaj
Veza između stepena i korena Negativni racionalni izložilac Decimalni izložilac Pravila stepenovanja za racionalne izložioce Svodjenje osnove na stepen prostog broja Koren iz korena Smena radi uprošćavanja Najvažnije formule
1. Veza između stepena i korena
Svaki stepen sa racionalnim izložiocem može se zapisati kao koren:
a 1 n = a n a^{\frac{1}{n}} = \sqrt[n]{a} a n 1 = n a
a m n = a m n = ( a n ) m a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m a n m = n a m = ( n a ) m
( 16 25 ) 1 2 = 16 25 = 16 25 = 4 5 \left(\frac{16}{25}\right)^{\frac{1}{2}} = \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5} ( 25 16 ) 2 1 = 25 16 = 25 16 = 5 4
8 1 3 = 8 3 = 2 8^{\frac{1}{3}} = \sqrt[3]{8} = 2 8 3 1 = 3 8 = 2
32 2 5 = 32 2 5 = ( 32 5 ) 2 = 2 2 = 4 32^{\frac{2}{5}} = \sqrt[5]{32^2} = \left(\sqrt[5]{32}\right)^2 = 2^2 = 4 3 2 5 2 = 5 3 2 2 = ( 5 32 ) 2 = 2 2 = 4
2. Negativni racionalni izložilac
Kombinujemo pravilo za negativan stepen sa definicijom racionalnog izložioca:
a − m n = 1 a m n = 1 a m n a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} = \frac{1}{\sqrt[n]{a^m}} a − n m = a n m 1 = n a m 1
Za razlomke kao osnovu, negativan izložilac okreće razlomak:
( a b ) − m n = ( b a ) m n \left(\frac{a}{b}\right)^{-\frac{m}{n}} = \left(\frac{b}{a}\right)^{\frac{m}{n}} ( b a ) − n m = ( a b ) n m
25 − 3 2 = 1 25 3 2 = 1 ( 5 2 ) 3 2 = 1 5 3 = 1 125 25^{-\frac{3}{2}} = \frac{1}{25^{\frac{3}{2}}} = \frac{1}{(5^2)^{\frac{3}{2}}} = \frac{1}{5^3} = \frac{1}{125} 2 5 − 2 3 = 2 5 2 3 1 = ( 5 2 ) 2 3 1 = 5 3 1 = 125 1
( 1 27 ) − 4 3 = 27 4 3 = ( 3 3 ) 4 3 = 3 4 = 81 \left(\frac{1}{27}\right)^{-\frac{4}{3}} = 27^{\frac{4}{3}} = (3^3)^{\frac{4}{3}} = 3^4 = 81 ( 27 1 ) − 3 4 = 2 7 3 4 = ( 3 3 ) 3 4 = 3 4 = 81
( 1 4 ) − 3 2 = 4 3 2 = ( 2 2 ) 3 2 = 2 3 = 8 \left(\frac{1}{4}\right)^{-\frac{3}{2}} = 4^{\frac{3}{2}} = (2^2)^{\frac{3}{2}} = 2^3 = 8 ( 4 1 ) − 2 3 = 4 2 3 = ( 2 2 ) 2 3 = 2 3 = 8
3. Decimalni izložilac
Decimalni izložilac uvek prvo pretvaramo u razlomak.
Izračunati 0,25 − 0,5 0{,}25^{-0{,}5} 0 , 2 5 − 0 , 5
0,25 = 1 4 , − 0,5 = − 1 2 0{,}25 = \frac{1}{4}, \qquad -0{,}5 = -\frac{1}{2} 0 , 25 = 4 1 , − 0 , 5 = − 2 1
( 1 4 ) − 1 2 = 4 1 2 = 4 = 2 \left(\frac{1}{4}\right)^{-\frac{1}{2}} = 4^{\frac{1}{2}} = \sqrt{4} = 2 ( 4 1 ) − 2 1 = 4 2 1 = 4 = 2
Izračunati ( 1 256 ) 0,375 \left(\dfrac{1}{256}\right)^{0{,}375} ( 256 1 ) 0 , 375
0,375 = 3 8 , 1 256 = 2 − 8 0{,}375 = \frac{3}{8}, \qquad \frac{1}{256} = 2^{-8} 0 , 375 = 8 3 , 256 1 = 2 − 8
( 2 − 8 ) 3 8 = 2 − 3 = 1 8 (2^{-8})^{\frac{3}{8}} = 2^{-3} = \frac{1}{8} ( 2 − 8 ) 8 3 = 2 − 3 = 8 1
4. Pravila stepenovanja za racionalne izložioce
Sva pravila koja važe za cele izložioce važe i za racionalne:
Pravilo Formula Množenje istih osnova a m n ⋅ a p q = a m n + p q a^{\frac{m}{n}} \cdot a^{\frac{p}{q}} = a^{\frac{m}{n} + \frac{p}{q}} a n m ⋅ a q p = a n m + q p Deljenje istih osnova a m n : a p q = a m n − p q a^{\frac{m}{n}} : a^{\frac{p}{q}} = a^{\frac{m}{n} - \frac{p}{q}} a n m : a q p = a n m − q p Stepen stepena ( a m n ) p q = a m n ⋅ p q \left(a^{\frac{m}{n}}\right)^{\frac{p}{q}} = a^{\frac{m}{n} \cdot \frac{p}{q}} ( a n m ) q p = a n m ⋅ q p Stepen proizvoda ( a b ) m n = a m n ⋅ b m n (ab)^{\frac{m}{n}} = a^{\frac{m}{n}} \cdot b^{\frac{m}{n}} ( ab ) n m = a n m ⋅ b n m
Uprostiti x 3 2 ⋅ y 4 5 ⋅ z 5 6 : ( x 5 4 ⋅ y 2 3 ⋅ z 7 12 ) x^{\frac{3}{2}} \cdot y^{\frac{4}{5}} \cdot z^{\frac{5}{6}} : (x^{\frac{5}{4}} \cdot y^{\frac{2}{3}} \cdot z^{\frac{7}{12}}) x 2 3 ⋅ y 5 4 ⋅ z 6 5 : ( x 4 5 ⋅ y 3 2 ⋅ z 12 7 )
x 3 2 − 5 4 ⋅ y 4 5 − 2 3 ⋅ z 5 6 − 7 12 = x 1 4 ⋅ y 2 15 ⋅ z 1 4 x^{\frac{3}{2} - \frac{5}{4}} \cdot y^{\frac{4}{5} - \frac{2}{3}} \cdot z^{\frac{5}{6} - \frac{7}{12}} = x^{\frac{1}{4}} \cdot y^{\frac{2}{15}} \cdot z^{\frac{1}{4}} x 2 3 − 4 5 ⋅ y 5 4 − 3 2 ⋅ z 6 5 − 12 7 = x 4 1 ⋅ y 15 2 ⋅ z 4 1
5. Svodjenje osnove na stepen prostog broja
Ovo je ključna tehnika za izračunavanje konkretnih vrednosti. Osnovu zapisujemo kao stepen prostog broja, a zatim množimo izložioce.
Kada vidiš broj kao osnovu, odmah proveri da li ga možeš zapisati kao stepen nekog manjeg broja: 4 = 2 2 4 = 2^2 4 = 2 2 8 = 2 3 8 = 2^3 8 = 2 3 27 = 3 3 27 = 3^3 27 = 3 3 32 = 2 5 32 = 2^5 32 = 2 5 64 = 2 6 64 = 2^6 64 = 2 6 243 = 3 5 243 = 3^5 243 = 3 5 1024 = 2 10 1024 = 2^{10} 1024 = 2 10
Izračunati ( 1 3 ) − 10 ⋅ 27 − 3 + 0,2 − 4 ⋅ 25 − 2 + ( 64 − 1 9 ) − 3 \left(\dfrac{1}{3}\right)^{-10} \cdot 27^{-3} + 0{,}2^{-4} \cdot 25^{-2} + (64^{-\frac{1}{9}})^{-3} ( 3 1 ) − 10 ⋅ 2 7 − 3 + 0 , 2 − 4 ⋅ 2 5 − 2 + ( 6 4 − 9 1 ) − 3
Sabirak 1: 3 10 ⋅ ( 3 3 ) − 3 = 3 10 ⋅ 3 − 9 = 3 1 = 3 3^{10} \cdot (3^3)^{-3} = 3^{10} \cdot 3^{-9} = 3^1 = 3 3 10 ⋅ ( 3 3 ) − 3 = 3 10 ⋅ 3 − 9 = 3 1 = 3
Sabirak 2: ( 1 5 ) − 4 ⋅ ( 5 2 ) − 2 = 5 4 ⋅ 5 − 4 = 5 0 = 1 \left(\frac{1}{5}\right)^{-4} \cdot (5^2)^{-2} = 5^4 \cdot 5^{-4} = 5^0 = 1 ( 5 1 ) − 4 ⋅ ( 5 2 ) − 2 = 5 4 ⋅ 5 − 4 = 5 0 = 1
Sabirak 3: ( 64 − 1 9 ) − 3 = 64 1 3 = ( 2 6 ) 1 3 = 2 2 = 4 (64^{-\frac{1}{9}})^{-3} = 64^{\frac{1}{3}} = (2^6)^{\frac{1}{3}} = 2^2 = 4 ( 6 4 − 9 1 ) − 3 = 6 4 3 1 = ( 2 6 ) 3 1 = 2 2 = 4
3 + 1 + 4 = 8 3 + 1 + 4 = 8 3 + 1 + 4 = 8
6. Koren iz korena
Pravilo za koren iz korena:
a m n = a n ⋅ m = a 1 n m \sqrt[n]{\sqrt[m]{a}} = \sqrt[n \cdot m]{a} = a^{\frac{1}{nm}} n m a = n ⋅ m a = a nm 1
Kada vidiš ugnježdene korene, odmah ih pretvori u stepene sa racionalnim izložiocima. Tada jednostavno množiš izložioce i sve spaja u jedan stepen.
Uprostiti a 3 ⋅ a 2 3 \sqrt[3]{\sqrt{a}} \cdot \sqrt{\sqrt[3]{a^2}} 3 a ⋅ 3 a 2
a 3 ⋅ 2 ⋅ a 2 2 ⋅ 3 = a 6 ⋅ a 2 6 = a 3 6 = a \sqrt[3 \cdot 2]{a} \cdot \sqrt[2 \cdot 3]{a^2} = \sqrt[6]{a} \cdot \sqrt[6]{a^2} = \sqrt[6]{a^3} = \sqrt{a} 3 ⋅ 2 a ⋅ 2 ⋅ 3 a 2 = 6 a ⋅ 6 a 2 = 6 a 3 = a
Uprostiti x x x x 3 ⋅ x x 3 x\sqrt{x\sqrt{x\sqrt[3]{x}}} \cdot \sqrt[3]{x\sqrt{x}} x x x 3 x ⋅ 3 x x
Prevodimo sve u stepene i sabiramo izložioce:
x 1 ⋅ x 1 2 ⋅ x 1 4 ⋅ x 1 12 ⋅ x 1 3 ⋅ x 1 6 = x 12 + 6 + 3 + 1 + 4 + 2 12 = x 28 12 = x 7 3 = x 2 x 3 x^1 \cdot x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} \cdot x^{\frac{1}{12}} \cdot x^{\frac{1}{3}} \cdot x^{\frac{1}{6}} = x^{\frac{12+6+3+1+4+2}{12}} = x^{\frac{28}{12}} = x^{\frac{7}{3}} = x^2\sqrt[3]{x} x 1 ⋅ x 2 1 ⋅ x 4 1 ⋅ x 12 1 ⋅ x 3 1 ⋅ x 6 1 = x 12 12 + 6 + 3 + 1 + 4 + 2 = x 12 28 = x 3 7 = x 2 3 x
7. Smena radi uprošćavanja
Kada se u izrazu pojave složeni izložioci, posebno ako se među njima uoči određena pravilnost ili veza, može se uvesti smena
za taj izložilac. Na taj način se izložilac pojednostavljuje, a ceo izraz prelazi u oblik koji je lakši za sređivanje.
Ideja je da izabereš smenu tako da svi stepenei postanu celi brojevi.
Ako imaš izraze x 1 / 4 x^{1/4} x 1/4 x 1 / 2 x^{1/2} x 1/2 x 3 / 4 x^{3/4} x 3/4 x x x 1 4 \frac{1}{4} 4 1
t = x 1 4 t=x^{\frac{1}{4}} t = x 4 1
Tada se sve lepo “pretvori”:
x 1 / 2 = ( x 1 / 4 ) 2 = t 2 x^{1/2} = (x^{1/4})^2 = t^2 x 1/2 = ( x 1/4 ) 2 = t 2
x 3 / 4 = ( x 1 / 4 ) 3 = t 3 x^{3/4} = (x^{1/4})^3 = t^3 x 3/4 = ( x 1/4 ) 3 = t 3
x = ( x 1 / 4 ) 4 = t 4 x = (x^{1/4})^4 = t^4 x = ( x 1/4 ) 4 = t 4
Uprostiti x − y x 3 / 4 + x 1 / 2 y 1 / 4 ⋅ x 1 / 2 y 1 / 4 + x 1 / 4 y 1 / 2 x 1 / 2 + y 1 / 2 ⋅ x 1 / 4 y 1 / 4 x 1 / 2 − 2 x 1 / 4 y 1 / 4 + y 1 / 2 \dfrac{x - y}{x^{3/4} + x^{1/2}y^{1/4}} \cdot \dfrac{x^{1/2}y^{1/4} + x^{1/4}y^{1/2}}{x^{1/2} + y^{1/2}} \cdot \dfrac{x^{1/4}y^{1/4}}{x^{1/2} - 2x^{1/4}y^{1/4} + y^{1/2}} x 3/4 + x 1/2 y 1/4 x − y ⋅ x 1/2 + y 1/2 x 1/2 y 1/4 + x 1/4 y 1/2 ⋅ x 1/2 − 2 x 1/4 y 1/4 + y 1/2 x 1/4 y 1/4
Uvedemo a = x 1 / 4 a = x^{1/4} a = x 1/4 b = y 1 / 4 b = y^{1/4} b = y 1/4 x = a 4 x = a^4 x = a 4 y = b 4 y = b^4 y = b 4 x − y = a 4 − b 4 x - y = a^4 - b^4 x − y = a 4 − b 4
( a 2 − b 2 ) ( a 2 + b 2 ) a 2 ( a + b ) ⋅ a b ( a + b ) a 2 + b 2 ⋅ a b ( a − b ) 2 = b 2 ( a + b ) a − b \frac{(a^2 - b^2)(a^2 + b^2)}{a^2(a + b)} \cdot \frac{ab(a + b)}{a^2 + b^2} \cdot \frac{ab}{(a - b)^2} = \frac{b^2(a + b)}{a - b} a 2 ( a + b ) ( a 2 − b 2 ) ( a 2 + b 2 ) ⋅ a 2 + b 2 ab ( a + b ) ⋅ ( a − b ) 2 ab = a − b b 2 ( a + b )
Vraćamo smenu: y 1 / 2 ( x 1 / 4 + y 1 / 4 ) x 1 / 4 − y 1 / 4 \dfrac{y^{1/2}(x^{1/4} + y^{1/4})}{x^{1/4} - y^{1/4}} x 1/4 − y 1/4 y 1/2 ( x 1/4 + y 1/4 )
Formula Napomena a 1 n = a n a^{\frac{1}{n}} = \sqrt[n]{a} a n 1 = n a Osnovna definicija a m n = a m n a^{\frac{m}{n}} = \sqrt[n]{a^m} a n m = n a m Opšti slučaj a − m n = 1 a m n a^{-\frac{m}{n}} = \dfrac{1}{\sqrt[n]{a^m}} a − n m = n a m 1 Negativni racionalni izložilac a m n = a n m \sqrt[n]{\sqrt[m]{a}} = \sqrt[nm]{a} n m a = nm a Koren iz korena ( a m n ) p q = a m p n q (a^{\frac{m}{n}})^{\frac{p}{q}} = a^{\frac{mp}{nq}} ( a n m ) q p = a n q m p Stepen stepena a m n ⋅ a p q = a m q + n p n q a^{\frac{m}{n}} \cdot a^{\frac{p}{q}} = a^{\frac{mq + np}{nq}} a n m ⋅ a q p = a n q m q + n p Množenje sa istom osnovom