Pangkat Bulat - UK 6 | Minat Matematika

Dalam pengerjaannya, bilangan berpangkat bulat positif, negatif, dan nol dapat diselesaikan dengan menggunakan sifat-sifat tertentu, yang kita sebut dengan sifat-sifat perpangkatan. Sifat-sifat ini dapat ditemukan saat kita mengerjakan soal-soal UK sebelumnya pada sub-bab pangkat bulat ini. Jika kalian belum mengerjakan soal-soal UK sebelumnya, silakan baca dan pelajari dulu melalui link di bawah:
Latihan Uji Kompetensi 1
Latihan Uji Kompetensi 2
Latihan Uji Kompetensi 3
Latihan Uji Kompetensi 4
Latihan Uji Kompetensi 5

Jika sudah selesai mempelajari UK-UK sebelumnya, sekarang saatnya mempelajari pembahasan soal UK 1.1.6 pada sub-bab pangkat bulat yang kami ambil dari buku PKS Matematika Peminatan kelas X oleh Wilson Simangunsong yang bisa kalian baca dan pelajari.

Sifat-sifat perpangkatan dinyatakan dengan teorema berikut ini:

\(a^m . a^n = a^{m+n} \)

\(\frac {a^m}{a^n} = a^{m-n} \)

\((a^m)^n = a^{mn} \)

\((a b)^m = a^m\ . \ b^m \)

Soal No. 1

\(\small 2^{-4}\ . \ 2^7\ = \ … \)

Pembahasan No. 1

\(\small 2^{-4}\ . \ 2^7 = 2^{-4+7} \)
\(\small \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 2^{3} \)

Soal No. 2

\(\small (-5)^{-7}\ . \ (-5)^9\ = \ …\)

Pembahasan No. 2

\(\small (-5)^{-7}\ . \ (-5)^9 = (-5)^{-7+9} \)
\(\small = (-5)^{2} \)
\(\small = (-5)\ . \ (-5) \)
\(\small = 25 \)

Soal No. 3

\(\small(6\ a^5\ b^{-3})\ . \ (2\ a^{-2}\ b^7)\ = \ …\)

Pembahasan No. 3

\(\small (6\ a^5\ b^{-3})\ . \ (2\ a^{-2}\ b^7) = 6\ . \ a^5\ . \ b^{-3}\ . \ 2\ . \ a^{-2}\ . \ b^7 \)
\(\small = 6\ . \ 2\ . \ a^5\ . \ a^{-2}\ . \ b^{-3}\ . \ b^7 \)
\(\small = 12\ . \ a^{5+(-2)}\ . \ b^{-3+7} \)
\(\small = 12\ . \ a^{3}\ . \ b^{4} \)

Soal No. 4

\(\small (16\ x^5\ y^3)\ . \left(\frac {1}{2^3}\ x^{-3}\ y^{-2}\right)\ = \ …\)

Pembahasan No. 4

\(\small (16\ x^5\ y^3)\ . \ \left(\frac {1}{2^3}\ x^{-3}\ y^{-2}\right) = 16\ . \ x^5\ . \ y^3\ . \ \frac {1}{2^3}\ . \ x^{-3}\ . \ y^{-2} \)
\(\small = 2^4\ . \ \frac {1}{2^3} . \ x^5\ . \ x^{-3} . \ y^3\ . \ y^{-2} \)
\(\small = \frac {2^4}{2^3} . \ x^5\ . \ x^{-3} . \ y^3\ . \ y^{-2} \)
\(\small = 2^{4-3} . \ x^{5+(-3)} . \ y^{3+(-2)} \)
\(\small = 2^1 . \ x^2 . \ y^1 \)
\(\small = 2\ x^2\ y \)

Soal No. 5

\(\small \left(2 – \frac 12 – \frac {1}{2^2} \right)^{-2}\ = \ …\)

Pembahasan No. 5

\(\small \left(2 - \frac 12 - \frac {1}{2^2} \right)^{-2} = \left(2 - \frac 12 - \frac {1}{4} \right)^{-2} \)
\(\small = \left(\frac 84 - \frac 24 - \frac {1}{4} \right)^{-2} \)
\(\small = \left(\frac {8-2-1}{4}\right)^{-2} \)
\(\small = \left(\frac {5}{4}\right)^{-2} \)
\(\small = \frac {5^{-2}}{4^{-2}} \)
\(\small = \frac {4^{2}}{5^{2}} \)
\(\small = \frac {16}{25} \)

Soal No. 6

\(\small a^9 : a^{-3}\ = \ … \)

Pembahasan No. 6

\(\small a^9 : a^{-3} = \frac {a^9}{a^-3} \)
\(\small = a^{9-(-3)} \)
\(\small = a^{9+3} \)
\(\small = a^{12} \)

Soal No. 7

\(\small \frac {3^{n+1}-3^{n-1}}{3^{n-1}-3^{n-2}}\ = \ … \)

Pembahasan No. 7

\(\small \frac {3^{n+1}-3^{n-1}}{3^{n-1}-3^{n-2}} = \frac {3^n\ . \ 3^1 - \frac {3^n}{3^1}}{\frac {3^n}{3^1}-\frac {3^n}{3^2}} \)
\(\small = \frac {3^n (3 - \frac {1}{3})}{3^n(\frac {1}{3}-\frac {1}{9})} \)
\(\small = \frac {3 - \frac {1}{3}}{\frac {1}{3}-\frac {1}{9}} \)
\(\small = \frac {3 - \frac {1}{3}}{\frac {1}{3}-\frac {1}{9}} \ . \frac 99 \)
\(\small = \frac {27 - 3}{3-1} \)
\(\small = \frac {24}{2} \)
\(\small = 12 \)

Soal No. 8

\(\small \frac {2^{-3}\ a^3\ b^{-4}}{\left(\frac 12\right)^4\ a^4\ b^{-5}}\ = \ … \)

Pembahasan No. 8

\(\small \frac {2^{-3}\ a^3\ b^{-4}}{\left(\frac 12\right)^4\ a^4\ b^{-5}} = \frac {2^{-3}\ a^3\ b^{-4}}{\frac {1^4}{2^4}\ a^4\ b^{-5}} \)
\(\small = \frac {2^{-3}\ a^3\ b^{-4}}{\frac {1}{2^4}\ a^4\ b^{-5}} \)
\(\small = \frac {2^{-3}\ 2^{4} \ a^3\ b^{-4}}{a^4\ b^{-5}} \)
\(\small = \frac {2^{-3+4} \ b^{-4-(-5)}}{a^{4-3}} \)
\(\small = \frac {2^{-3+4} \ b^{-4+5}}{a^{4-3}} \)
\(\small = \frac {2^{1} \ b^{1}}{a^{1}} \)
\(\small = \frac {2b}{a} \)

Soal No. 9

\(\small \left(\left(\frac 12 \right)^3 \right)^{-2}\ = \ … \)

Pembahasan No. 9

\(\small = \left(\frac 12 \right)^{3 . \ (-2)} \)
\(\small = \left(\frac 12 \right)^{-6} \)
\(\small = \frac {1^{-6}}{2^{-6}} \)
\(\small = \frac {2^{6}}{1^{6}} \)
\(\small = \frac {64}{1} \)
\(\small = 64 \)

Soal No. 10

\(\small (2\ . \ (2^2 \ . \ (2^3)^{-1})^{-2})\ = \ … \)

Pembahasan No. 10

\(\small (2\ . \ (2^2 \ . \ (2^3)^{-1})^{-2}) = (2\ . \ (2^2 \ . \ 2^{3\ . \ -1})^{-2}) \)
\(\small = (2\ . \ (2^2 \ . \ 2^{-3})^{-2}) \)
\(\small = (2\ . \ 2^{2\ . \ -2} \ . \ 2^{-3\ . \ -2}) \)
\(\small = (2\ . \ 2^{-4} \ . \ 2^{6}) \)
\(\small = 2^1\ . \ 2^{-4} \ . \ 2^{6} \)
\(\small = 2^{1+(-4)+6} \)
\(\small = 2^{3} \)
\(\small = 8 \)

Soal No. 11

\(\small \left(\left(\frac 12 \right)^2 \ . \ 2^2 \right)^{-2}\ = \ … \)

Pembahasan No. 11

\(\small \left(\left(\frac 12 \right)^2 \ . \ 2^2 \right)^{-2} = \left(\frac {1^2}{2^2} \ . \ 2^2 \right)^{-2} \)
\(\small = \left(\frac {1}{4} \ . \ 4 \right)^{-2} \)
\(\small = 1^{-2} \)
\(\small = 1 \)

Soal No. 12

\(\small (2\ a^3\ b^{-4})^2\ = \ … \)

Pembahasan No. 12

\(\small (2\ a^3\ b^{-4})^2 = 2^2\ a^{3\ . \ 2}\ b^{-4\ . \ 2} \)
\(\small = 2^2\ a^{6}\ b^{-8} \)
\(\small = \frac {4\ a^{6}}{b^8} \)

Soal No. 13

\(\small \frac {(0,6)^0 – (0,1)^{-1}}{\left( \frac {3}{2^3} \right)^{-1} \left( \frac 32\right)^3 + \left( – \frac 13\right)^{-1}}\ = \ … \)

Pembahasan No. 13

\(\small \frac {(0,6)^0 - (0,1)^{-1}}{\left( \frac {3}{2^3} \right)^{-1} \left( \frac 32\right)^3 + \left( - \frac 13\right)^{-1}} = \frac {\left( \frac {6}{10} \right)^{0} - \left( \frac {1}{10} \right)^{-1}}{\left( \frac {3}{2^3} \right)^{-1} \left( \frac 32\right)^3 + \left( - \frac 13\right)^{-1}} \)
\(\small = \frac {1 - \frac {1^{-1}}{10^{-1}}}{\frac {3^{-1}}{2^{-3}} \frac {3^3}{2^3} + \left( - \frac {1^{-1}}{3^{-1}} \right)} \)
\(\small = \frac {1 - \frac {10^1}{1^1}}{\frac {2^3}{3^1} \frac {3^3}{2^3} + \left( - \frac {3^1}{1^1} \right)} \)
\(\small = \frac {1 - \frac {10^1}{1^1}}{\frac {3^3}{3^1} - \frac {3^1}{1^1}} \)
\(\small = \frac {1 - \frac {10}{1}}{\frac {27}{3} - \frac {3}{1}} \)
\(\small = \frac {1 - 10}{9 - 3} \)
\(\small = \frac {-9}{6} \)
\(\small = -\frac {9}{6} \)
\(\small = -\frac {3}{2} \)

Soal No. 14

\(\small \left( \frac {3 \ a^6\ . \ b^{-5}}{81 \ a^9\ . \ b^{-2}} \right)^{-1}\ = \ … \)

Pembahasan No. 14

\(\small \left( \frac {3 \ a^6\ . \ b^{-5}}{81 \ a^9\ . \ b^{-2}} \right)^{-1} = \frac {81 \ a^9\ . \ b^{-2}}{3 \ a^6\ . \ b^{-5}} \)
\(\small = \frac {3^4 \ a^9\ . \ b^{-2}}{3^1 \ a^6\ . \ b^{-5}} \)
\(\small = 3^{4-1} \ a^{9-6}\ . \ b^{-2-(-5)} \)
\(\small = 3^{4-1} \ a^{9-6}\ . \ b^{-2 + 5)} \)
\(\small = 3^3 \ a^3\ . \ b^3 \)
\(\small = (3ab)^3 \)

Soal No. 15

\(\small \frac {x^{-1} + y^{-1}}{x^{-1} – y^{-1}}\ = \ … \)

Pembahasan No. 15

\(\small \frac {x^{-1} + y^{-1}}{x^{-1} - y^{-1}} = \frac {\frac {1}{x} + \frac {1}{y}}{\frac {1}{x} - \frac {1}{y}} \)
\(\small = \frac {\frac {y}{xy} + \frac {x}{xy}}{\frac {y}{xy} - \frac {x}{xy}} \)
\(\small = \frac {\frac {x+y}{xy}}{\frac {x-y}{xy}} \)
\(\small = {\frac {x+y}{xy}} : {\frac {x-y}{xy}} \)
\(\small = {\frac {x+y}{xy}}\ . \ {\frac {xy}{x-y}} \)
\(\small = {\frac {x+y}{x-y}} \)

Itu dia pembahasan soal Latihan Uji Kompetensi 6 atau UK 1.1.6 pada sub-bab pangkat bulat yang kami ambil dari buku PKS Matematika Peminatan kelas X oleh Wilson Simangunsong.

Untuk pembahasan soal lainnya bisa kalian cek Paket Soal Lain.

Apabila ada hal yang ingin disampaikan silakan komentar di kolom komentar di bawah.

Jangan berhenti belajar dan mencoba hal baru, bagikan pembahasan soal dari kami ke teman-temanmu agar mereka juga tahu dan bisa ikut belajar bersama kami.

TERIMA KASIH…

Pangkat Bulat – UK 6