Sifat-Sifat Bilangan Berpangkat

Beberapa sifat-sifat perpangkatan pada perkalian, pembagian, pangkat nol dan pangkat negatif.

am×an =am+na^m\times a^n\ = a^{m+n}amam=am−n\large\frac{a^m}{a^m} \normalsize\hspace2.4em = a^{m-n}(am)n=am×n(a^m)^n \hspace1.2em= a^{m\times n}(a×b)m=am×bm(a\times b)^m = a^m\times b^m(ab)m=ambm\large(\frac{a}{b})^m \normalsize\hspace1.3em= \large\frac{a^m}{b^m}a0=1 ,a̸=0a^0 \hspace2.7em = 1\ ,\enspace a\not = 0a−m=1ama^{-m} \hspace1.9em =\large\frac{1}{a^m}

Penjelasan masing-masing sifat bilangan berpangkat.

am×an=am+na^m\times a^n = a^{m+n}

am×an=a×a×a×a⎵m×a×a×a⎵na^m\times a^n = \underbrace{a\times a\times a\times a}_{m}\times \underbrace{a\times a\times a}_{n}=am+n\hspace3.4em = a^{m+n}Contoh 1:74×73=74+3=777^4\times 7^3=7^{4+3}=7^7Contoh 2:24×7(24)=24(1+7)2^4\times 7(2^4)=2^4(1+7)=24×8\hspace4.35em =2^4\times 8=24×23\hspace4.35em =2^4\times 2^3=24+3\hspace4.35em =2^{4+3}=27\hspace4.35em =2^7

amam=am−n\large\frac{a^m}{a^m} \normalsize = a^{m-n}

amam=a×a×a×a×a⏞ma×a×a⎵n=a×a×a×a×aa×a×a\large\frac{a^m}{a^m} \normalsize = \large\frac{\overbrace{a\times a\times a\times a\times a}^{m}}{\underbrace{a\times a\times a}_{n}} \normalsize = \large\frac{\cancel a\times \cancel a\times \cancel a\times a\times a}{\cancel a\times \cancel a\times \cancel a}=am−n\hspace1.25em =a^{m-n}Contoh :3533=35−3=32\large\frac{3^5}{3^3}\normalsize = 3^{5-3} = 3^2

(am)n=am×n(a^m)^n = a^{m\times n}

(am)n=(a×a×a⎵m)n=(a×a×a⎵m)×(a×a×a⎵m)⎵n(a^m)^n = (\underbrace{a\times a\times a}_{m})^n = \underbrace{(\underbrace{a\times a\times a}_{m})\times (\underbrace{a\times a\times a}_{m})}_{n}=am×n\hspace9.1em =a^{m\times n}Contoh :(43)2=43×2=46(4^3)^2=4^{3\times 2}=4^6

(a×b)m=am×bm(a\times b)^m = a^m\times b^m

(a×b)m=(a×b)×(a×b)×(a×b)⎵m(a\times b)^m = \underbrace{(a\times b)\times (a\times b)\times (a\times b)}_{m}=a×a×a×b×b×b\hspace3.6em = a\times a\times a\times b\times b\times b=am×bm\hspace3.6em = a^m\times b^mContoh :(4×5)3=43×53(4\times 5)^3 = 4^3\times 5^3

(ab)m=ambm\large(\frac{a}{b})^m \normalsize = \large\frac{a^m}{b^m}

(ab)m=(ab)×(ab)×(ab)⎵m=a×a×ab×b×b\large(\frac{a}{b})^m \normalsize = \underbrace{(\frac{a}{b})\times (\frac{a}{b})\times (\frac{a}{b})}_{m} = \large\frac{a\times a\times a}{b\times b\times b}=ambm\hspace2.4em = \large\frac{a^m}{b^m}Contoh :(52)3=5323\large(\frac{5}{2})^3\normalsize = \large\frac{5^3}{2^3}

a0=1a^0 = 1

pembagian berpangkat2323=23−3=20\large\frac{2^3}{2^3}\normalsize =2^{3-3}=2^0hasil pembagian2323=88=1\large\frac{2^3}{2^3}\normalsize =\large\frac{8}{8} \normalsize=1

Jadi dapat disimpulkan bahwa :

20=12^0 = 1

a−m=1ama^{-m}=\large\frac{1}{a^m}

a−m=a0−m=a0am=1ama^{-m}=a^{0-m}=\large\frac{a^0}{a^m}\normalsize = \large\frac{1}{a^m}Contoh :2−4=124=1162^{-4}=\large\frac{1}{2^4} \normalsize = \large\frac{1}{16}
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