10.8: Integer Exponents na Nukuu ya kisayansi (Sehemu ya 1)

Malengo ya kujifunza

• Tumia ufafanuzi wa exponent hasi
• Kurahisisha maneno na exponents integer
• Badilisha kutoka notation decimal kwa notation kisayansi
• Badilisha notation ya kisayansi kwa fomu ya decimal
• Kuzidisha na ugawanye kwa kutumia nukuu

kuwa tayari!

Kabla ya kuanza, fanya jaribio hili la utayari.

1. ni thamani ya mahali ya 6 katika idadi 64,891 nini? Kama amekosa tatizo hili, mapitio Mfano 1.1.3.
2. Jina la decimal 0.0012. Ikiwa umekosa tatizo hili, tathmini Zoezi 5.1.1.
3. Ondoa: 5 - (-3). Ikiwa umekosa tatizo hili, kagua Mfano 3.5.8.

Tumia Ufafanuzi wa Mtazamo Mbaya

Mali ya Quotient ya Exponents, ilianzishwa katika Gawanya Monomials, ilikuwa na aina mbili kulingana na kama exponent katika nambari au denominator ilikuwa kubwa.

Ufafanuzi: Mali ya Quotient ya Watazamaji

Ikiwa ni namba halisi, a - 0, na m, n ni namba nzima, basi

\[\dfrac{a^{m}}{a^{n}} = a^{m-n},\; m>n \quad and \quad \dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n-m}},\; n>m\]

Nini kama sisi tu Ondoa exponents, bila kujali ambayo ni kubwa? Hebu fikiria\(\dfrac{x^{2}}{x^{5}}\). Sisi Ondoa exponent katika denominator kutoka exponent katika nambari.

\[\begin{split} &\; \dfrac{x^{2}}{x^{5}} \\ &x^{2-5} \\ &x^{-3} \end{split}\]

Tunaweza pia kurahisisha\(\dfrac{x^{2}}{x^{5}}\) kwa kugawa mambo ya kawaida:\(\dfrac{x^{2}}{x^{5}}\).

\[\begin{split} &\dfrac{\cancel{x} \cdot \cancel{x}}{\cancel{x} \cdot \cancel{x} \cdot x \cdot x \cdot x} \\ &\qquad \quad \dfrac{1}{x^{3}} \end{split}\]

Hii ina maana kwamba\(x^{-3} = \dfrac{1}{x^{3}}\) na inatuongoza kwa ufafanuzi wa exponent hasi.

Ufafanuzi: exponent hasi

Ikiwa n ni integer chanya na ± 0, basi\(a^{−n} = \dfrac{1}{a^{n}}\).

Mtazamaji hasi anatuambia kuandika tena maneno kwa kuchukua usawa wa msingi na kisha kubadilisha ishara ya exponent. Maneno yoyote ambayo ina vielelezo hasi hayakufikiriwa kuwa katika fomu rahisi. Sisi kutumia ufafanuzi wa exponent hasi na mali nyingine ya exponents kuandika kujieleza na exponents chanya tu.

Mfano\(\PageIndex{1}\):

Kurahisisha: (a^{) 4 -2} (b) ^{10 1-3}

Suluhisho

(a) 4 ^-2

Tumia ufafanuzi wa exponent hasi,\(a^{−n} = \dfrac{1}{a^{n}}\).	$$\ dfrac {1} {4^ {2}} $
Kurahisisha.	$$\ dfrac {1} {16} $$

(b) ^{10 1-3}

Tumia ufafanuzi wa exponent hasi,\(a^{−n} = \dfrac{1}{a^{n}}\).	$$\ dfrac {1} {10^ {3}} $
Kurahisisha.	$$\ dfrac {1} {1000} $$

Zoezi\(\PageIndex{1}\):

Kurahisisha: (a^{) 2 —3} (b) 10 ^-2

Jibu

\(\frac{1}{8}\)

Jibu b

\(\frac{1}{100}\)

Zoezi\(\PageIndex{2}\):

Kurahisisha: (a^{) 3 -2} (b) ^{10 -4}

Jibu

\(\frac{1}{9}\)

Jibu b

\(\frac{1}{10,000}\)

Wakati kurahisisha kujieleza yoyote na exponents, ni lazima kuwa makini kwa usahihi kutambua msingi kwamba ni alimfufua kwa kila exponent.

Mfano\(\PageIndex{2}\):

Kurahisisha: (a) (-3^{) -2} (b) -3 ^—2

Suluhisho

Hasi katika exponent haiathiri ishara ya msingi.

a) (-3) ^-2

Mtazamo hutumika kwa msingi, -3.	$$ (-3) ^ {-2} $$
Chukua usawa wa msingi na ubadilishe ishara ya mtangazaji.	$$\ dfrac {1} {(-3) ^ {2}} $
Kurahisisha.	$$\ dfrac {1} {9} $$

(b) -3 ^—2

Maneno -3 ^—2 inamaanisha “pata kinyume cha 3 ^—2. exponent inatumika tu kwa msingi, 3.	$-3^ {-2} $$
Andika upya kama bidhaa na -1.	$-1\ cdot 3^ {-2} $
Chukua usawa wa msingi na ubadilishe ishara ya mtangazaji.	$-1\ dot\ dot\ drac {1} {3^ {2}} $
Kurahisisha.	$$-\ dfrac {1} {9} $$

Zoezi\(\PageIndex{3}\):

Kurahisisha: (a) (-5^{) -2} (b) -5 ^-2

Jibu

\(\frac{1}{25}\)

Jibu b

\(-\frac{1}{25}\)

Zoezi\(\PageIndex{4}\):

Kurahisisha: (a) (^{-2) -2} (b) ^{-1 -2}

Jibu

\(\frac{1}{4}\)

Jibu b

\(-\frac{1}{4}\)

Lazima tuwe makini kufuata utaratibu wa shughuli. Katika mfano unaofuata, sehemu (a) na (b) zinaonekana sawa, lakini tunapata matokeo tofauti.

Mfano\(\PageIndex{3}\):

Kurahisisha: (a) 4 • 2 ^-1 (b) (4 • 2^{) -1}

Suluhisho

Kumbuka daima kufuata utaratibu wa shughuli.

(a) 4 • ^{2 -1}

Je exponents kabla ya kuzidisha.	$4\ dot 2^ {-1} $
Tumia\(a^{−n} = \dfrac{1}{a^{n}}\).	$4\ dot\ dot\ drac {1} {2} {1}} $
Kurahisisha.	$2$$

(b) (4 • 2^{) -1}

Kurahisisha ndani ya mabano kwanza.	$$ (8) ^ {-1} $$
Tumia\(a^{−n} = \dfrac{1}{a^{n}}\).	$$\ dfrac {1} {8^ {1}} $
Kurahisisha.	$$\ dfrac {1} {8} $$

Zoezi\(\PageIndex{5}\):

Kurahisisha: (a) 6 • 3 ^-1 (b) (6 • 3^{) -1}

Jibu

\(2\)

Jibu b

\(\frac{1}{18}\)

Zoezi\(\PageIndex{6}\):

Kurahisisha: (a) 8 • ^{2 -2} (b) (8 • ^{2) -1}

Jibu

\(2\)

Jibu b

\(\frac{1}{256}\)

Wakati variable ni alimfufua kwa exponent hasi, sisi kuomba ufafanuzi njia ile ile tulivyofanya na idadi.

Mfano\(\PageIndex{4}\):

Kurahisisha: ^{x -6}.

Suluhisho

Tumia ufafanuzi wa exponent hasi,\(a^{−n} = \dfrac{1}{a^{n}}\).

$$\ dfrac {1} {x^ {6}} $

Zoezi\(\PageIndex{7}\):

Kurahisisha: ^{y -7}.

Jibu

\(\frac{1}{y^7}\)

Zoezi\(\PageIndex{8}\):

Kurahisisha: z ^-8.

Jibu

\(\frac{1}{z^8}\)

Wakati kuna bidhaa na exponent tunapaswa kuwa makini kutumia exponent kwa kiasi sahihi. Kwa mujibu wa utaratibu wa shughuli, maneno katika mabano yanarahisishwa kabla ya watazamaji kutumiwa. Tutaona jinsi hii inafanya kazi katika mfano unaofuata.

Mfano\(\PageIndex{5}\):

Kurahisisha: (a) 5y ^-1 (b) (^{5y) -1} (c) (-5y^{) -1}

Suluhisho

(a) ^{5y -1}

Taarifa exponent inatumika kwa tu y msingi.	$5y^ {-1} $$
Chukua usawa wa y na ubadilishe ishara ya mtangazaji.	$5\ dot\ dot\ drac {1} {y^ {1}} $
Kurahisisha.	$$\ dfrac {5} {y} $$

(b) (5y^{) -1}

Hapa mabano hufanya exponent kuomba 5y msingi.	$$ (5y) ^ {-1} $$
Chukua usawa wa 5y na ubadilishe ishara ya mtangazaji.	$$\ dfrac {1} {(5y) ^ {1}} $
Kurahisisha.	$$\ dfrac {1} {5y} $$

(c) (-5y^{) -1}

Msingi ni -5y. Chukua usawa wa -5y na ubadili ishara ya mtangazaji.	$$\ dfrac {1} {(-5y) ^ {1}} $
Kurahisisha.	$$\ dfrac {1} {-5y} $$
Tumia\(\dfrac{a}{-b} = - \dfrac{a}{b}\).	$$-\ dfrac {1} {5y} $$

Zoezi\(\PageIndex{9}\):

Kurahisisha: (a) 8p ^-1 (b) (^{8p) -1} (c) (-8p^{) -1}

Jibu

\(\frac{8}{p}\)

Jibu b

\(\frac{1}{8p}\)

Jibu c

\(-\frac{1}{8p}\)

Zoezi\(\PageIndex{10}\):

Kurahisisha: (a) 11q ^-1 (b) (^{11q) -1} (c) (-11q^{) -1}

Jibu

\(\frac{11}{q}\)

Jibu b

\(\frac{1}{11q}\)

Jibu c

\(-\frac{1}{11q}\)

Sasa kwa kuwa tuna defined exponents hasi, Quotient Mali ya Exponents mahitaji aina moja tu\(\dfrac{a^{m}}{a^{n}} = a^{m − n}\), ambapo ± 0 na m na n ni integers.

Wakati exponent katika denominator ni kubwa kuliko exponent katika nambari, exponent ya quotient itakuwa hasi. Kama matokeo inatupa exponent hasi, sisi kuandika upya kwa kutumia ufafanuzi wa exponents hasi,\(a^{−n} = \dfrac{1}{a^{n}}\).

Kurahisisha Maneno na Exponents Integer

Mali zote exponent sisi maendeleo mapema katika sura hii na idadi nzima exponents kuomba exponents integer, pia. Tunawapa tena hapa kwa kumbukumbu.

Muhtasari wa Mali Exponent

Ikiwa a, b ni namba halisi na m, n ni integers, basi

Bidhaa Mali	^m • ^n = ^m + n
Power Mali	(^m) ^n = ^m • n
Bidhaa kwa Mali ya Nguvu	(ab) ^m = ^m b ^m
Mali ya Quotient	\(\dfrac{a^{m}}{a^{n}}\)= ^m - n, a - 0, m> n
	\(\dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n-m}}\), a - 0,0, n> m
Zero Exponent Mali	a ^0 = 1, a ∙ 0
Quotient kwa Mali Nguvu	\(\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}\), b - 11,0
Ufafanuzi wa Mtazamo Mbaya	\(a^{-n} = \dfrac{1}{a^{n}}\)

Mfano\(\PageIndex{6}\):

Kurahisisha: (a) x ^-4 • x ⁶ (b) ^{y -6} • y ⁴ (c) ^{z -5} • ^{z -3}

Suluhisho

(a) ^{x -4} • x ⁶

Tumia Mali ya Bidhaa, ^m • ^n = ^m + n.	$x^ {-4+6} $$
Kurahisisha.	$x^ {2} $$

(b) y ^-6 • y ⁴

Msingi ni sawa, hivyo kuongeza exponents.	$$y ^ {-6+4} $$
Kurahisisha.	$y^ {-2} $$
Tumia ufafanuzi wa exponent hasi,\(a^{−n} = \dfrac{1}{a^{n}}\).	$$\ dfrac {1} {y^ {2}} $

(c) ^{z -5} • ^{z -3}

Msingi ni sawa, hivyo kuongeza exponents.	$z^ {-5-3} $$
Kurahisisha.	$z^ {-8} $$
Tumia ufafanuzi wa exponent hasi,\(a^{−n} = \dfrac{1}{a^{n}}\).	$$\ dfrac {1} {z^ {8}} $

Zoezi\(\PageIndex{11}\):

Kurahisisha: (a) x ^-3 • x ⁷ (b^{) y -7} • y ² (c) ^{z -4} • z ^-5

Jibu

\(x^4\)

Jibu b

\(\frac{1}{y^5}\)

Jibu c

\(\frac{1}{z^9}\)

Zoezi\(\PageIndex{12}\):

Kurahisisha: a ^-1 • a ⁶ (b) b ^-6 • b ⁴ (c) c ^-8 • c ^-7

Jibu

\(a^5\)

Jibu b

\(\frac{1}{b^4}\)

Jibu c

\(\frac{1}{c^{15}}\)

Katika mifano miwili ijayo, tutaweza kuanza kwa kutumia Mali Commutative kwa kundi vigezo sawa pamoja. Hii inafanya kuwa rahisi kutambua besi kama kabla ya kutumia Bidhaa Mali ya Exponents.

Mfano\(\PageIndex{7}\):

Rahisisha: (m ^{4 n -3}) (^{m -5} n ^-2).

Suluhisho

Matumizi Mali Commutative kupata kama besi pamoja.	$$m^ {4} m^ {-5}\ cdot n^ {-2} n^ {-3} $$
Kuongeza exponents kwa kila msingi.	$$ m^ {-1}\ cdot n^ {-5} $$
Chukua usawa na ubadilishe ishara za wafuasi.	$$\ dfrac {1} {m^ {1}}\ dot\ dfrac {1} {n^ {5}} $
Kurahisisha.	$$\ dfrac {1} {mn^ {5}} $

Zoezi\(\PageIndex{13}\):

Rahisisha: (p ^{6 q -1}) (^{p -9 q -1}).

Jibu

\(\frac{1}{p^3q^3}\)

Zoezi\(\PageIndex{14}\):

Rahisisha: (r ^{5 s -3}) (^{r -7} s ^-5).

Jibu

\(\frac{1}{r^2 s^8}\)

Kama monomials na coefficients namba, sisi kuzidisha coefficients, kama tulivyofanya katika Matumizi kuzidisha Mali ya Exponents.

Mfano\(\PageIndex{8}\):

Rahisisha: (^{2x -6} y ⁸) (-5x ^{5 y -3}).

Suluhisho

Andika upya na besi kama pamoja.	$2 (-5)\ cdot (x ^ {-6} x ^ {5})\ cdot (y ^ {8} y ^ {-3}) $$
Kurahisisha.	$-10\ cdot x ^ {-1}\ cdot y ^ {5} $
Tumia ufafanuzi wa exponent hasi,\(a^{−n} = \dfrac{1}{a^{n}}\).	$-10\ dot\ dot\ drac {1} {x^ {1}}\ dot ^ {5} $
Kurahisisha.	$$\ dfrac {-10y^ {5}} {x} $$

Zoezi\(\PageIndex{15}\):

Rahisisha: (^{3u -5} v ⁷) (-4u ⁴ v ^-2).

Jibu

\(-\frac{12v^5}{u}\)

Zoezi\(\PageIndex{16}\):

Rahisisha: (^{-6c -6} d ⁴) (^{-5c -2 d -1}).

Jibu

\(\frac{30d^3}{c^8}\)

Katika mifano miwili ijayo, tutatumia Mali ya Nguvu na Bidhaa kwa Mali ya Nguvu.

Mfano\(\PageIndex{9}\):

Kurahisisha: (k ^{3) -2}.

Suluhisho

Tumia Bidhaa kwa Mali ya Nguvu, (ab) ^m = ^m b ^m.	$$k^ {3 (-2)} $$
Kurahisisha.	$$k^ {-6} $$
Andika upya na exponent chanya.	$$\ dfrac {1} {k^ {6}} $

Zoezi\(\PageIndex{17}\):

Kurahisisha: (x ^{4) -1}.

Jibu

\(\frac{1}{x^4}\)

Zoezi\(\PageIndex{18}\):

Kurahisisha: (y ^{2) -2}.

Jibu

\(\frac{1}{y^4}\)

Mfano\(\PageIndex{10}\):

Kurahisisha: (^{5x 1-3}) ².

Suluhisho

Tumia Bidhaa kwa Mali ya Nguvu, (ab) ^m = ^m b ^m.	$5^ {2} (x^ {-3}) ^ {2} $$
Kurahisisha 5 ^2 na kuzidisha exponents ya x kutumia Power Mali, (^m) ^n = ^m • n.	$25k ^ {-6} $$
Andika upya x ^-6 kwa kutumia ufafanuzi wa exponent hasi,\(a^{−n} = \dfrac{1}{a^{n}}\).	$25\ dot\ dot\ drac {1} {x^ {6}} $
Kurahisisha.	$$\ dfrac {25} {x^ {6}} $

Zoezi\(\PageIndex{19}\):

Kurahisisha: (^{8a -4}) ².

Jibu

\(\frac{64}{a^8}\)

Zoezi\(\PageIndex{20}\):

Kurahisisha: (^{2c -4}) ³.

Jibu

\(\frac{8}{c^12}\)

Ili kurahisisha sehemu, tunatumia Mali ya Quotient.

Mfano\(\PageIndex{11}\):

Kurahisisha:\(\dfrac{r^{5}}{r^{−4}}\).

Suluhisho

Tumia Mali ya Quotient,\(\dfrac{a^{m}}{a^{n}} = a^{m-n}\).	$$r^ {5- (\ textcolor {nyekundu} {-4})} $$
Kuwa makini kuondoa 5 - (\ textcolor {nyekundu} {-4}).
Kurahisisha.	$$r^ {9} $$

Zoezi\(\PageIndex{21}\):

Kurahisisha:\(\dfrac{x^{8}}{x^{−3}}\).

Jibu

\(x^{11}\)

Zoezi\(\PageIndex{22}\):

Kurahisisha:\(\dfrac{y^{7}}{y^{-6}}\).

Jibu

\(y^{13}\)

Wachangiaji na Majina

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