Jawaban: B
\(\left(\dfrac{4a^2b^3}{2ab^2}\right)^2\)
\([2a^{(2\:-\:1)}b^{(3\:-\:2)}]^2\)
\((2ab)^2\)
\(2^2a^2b^2\)
\(4a^2b^2\)
Hasil dari \(\dfrac{2^{-1} + 2^0\:-\:2^{-2}}{\left(\dfrac{2}{5}\right)^{-1}}\) adalah …
Jawaban: A
\(\dfrac{\dfrac{1}{2} + 1\:-\:\dfrac{1}{2^2}}{\left(\dfrac{5}{2}\right)^{+1}}\)
\(\dfrac{\dfrac{1}{2} + 1\:-\:\dfrac{1}{4}}{\dfrac{5}{2}}\)
\(\dfrac{\dfrac{2}{4} + \dfrac{4}{4}\:-\:\dfrac{1}{4}}{\dfrac{5}{2}}\)
\(\dfrac{\dfrac{2 + 4 \:-\: 1}{4}}{\dfrac{5}{2}}\)
\(\dfrac{\dfrac{5}{4}}{\dfrac{5}{2}}\)
\(\dfrac{5}{4}\times {\dfrac{2}{5}}\)
\(\dfrac{10}{20}\)
\(\dfrac{1}{2}\)
Bentuk sederhana dari \(\left(\dfrac{x^{-2}y^{2}}{z^{-1}}\right)^{-2} \div \left(\dfrac{x^{-1}y}{z}\right)^2\) adalah …
Jawaban: C
\(\dfrac{x^4 y^{-4}}{z^2} \div \dfrac{x^{-2}y^2}{z^2}\)
\(\dfrac{x^4 y^{-4}}{\cancel{z^2}} \times \dfrac{\cancel{z^2}}{x^{-2}y^2}\)
\(x^{[4\:-\:(-2)]}\cdot y^{(-4\:-\:2)}\)
\(x^6 \cdot y^{-6}\)
\(\dfrac{x^6}{y^6}\)
Jawaban: C
\(\dfrac{2^{21 + 3} \:-\: 2^{21}}{2^{21}}\)
\(\dfrac{2^{21}\cdot 2^3 \:-\: 2^{21}}{2^{21}}\)
\(\dfrac{2^{21}\cdot 8 \:-\: 2^{21}\cdot 1}{2^{21}}\:\:\:\:\:\color{blue}\text{faktorkan}\)
\(\dfrac{\cancel{2^{21}}(8\:-\:1)}{\cancel{2^{21}}}\)
\(8\:-\:1\)
\(7\)
Bentuk sederhana dari \(\dfrac{x^{-2}\:-\:y^{-2}}{\dfrac{1}{x}\:-\:\dfrac{1}{y}}\) adalah …
Jawaban: A
\(\dfrac{\dfrac{y^2 \:-\:x^2}{x^2y^2}}{\dfrac{y\:-\:x}{xy}}\)
\(\dfrac{y^2 \:-\:x^2}{xy\cdot \cancel{xy}}\times \dfrac{\cancel{xy}}{y\:-\:x}\)
\(\dfrac{(y + x)(\cancel{y\:-\:x)}}{xy}\times \dfrac{1}{\cancel{y\:-\:x}}\)
\(\dfrac{y+x}{xy}\)
Jika \(\dfrac{120^2 \times 135^3}{50^4} = 2^a \times 3^b \times 5^c\), maka nilai \(2a + b \: – \: c\) adalah …
Jawaban: B
Tentukan dahulu faktorisasi prima dari 120, 135, dan 50
Faktorisasi prima dari 120 = \(2^3 \times 3 \times 5\)
Faktorisasi prima dari 135 = \(3^3 \times 5\)
Faktorisasi prima dari 50 = \(2 \times 5^2\)
\(\dfrac{120^2 \times 135^3}{50^4} = \dfrac{(2^3 \cdot 3 \cdot 5)^2 \times (3^3 \cdot 5)^3}{(2 \times 5^2)^4}\)
\(\dfrac{2^6\cdot 3^2 \cdot 5^2 \cdot 3^9 \cdot 5^3}{2^4 \cdot 5^8}\)
\(\dfrac{2^6\cdot 3^{2 + 9} \cdot 5^{2 + 3}}{2^4 \cdot 5^8}\)
\(2^{6 – 4}\cdot 3^{11} \cdot 5^{5-8}\)
\(2^2\cdot 3^{11} \cdot 5^{-3}\)
\(2^2\times 3^{11} \times 5^{-3} = 2^a \times 3^b \times 5^c\)
Dengan menyamakan pangkat bilangan-bilangan pokok ruas kiri dengan ruas kanan, diperoleh:
\(a = 2\)
\(b = 11\)
\(c = -3\)
Sehingga,
\(2a + b \: – \: c = 2(2) + 11 – (-3)\)
\(= 4 + 11 + 3\)
\(= 18\)
Urutan \(10^{300};\:\:\:\:21^{200};\:\:\:35^{100};\:\:\:2^{500}\) dari yang terkecil sampai terbesar adalah …
Jawaban: C
Ubah semua bilangan menjadi perpangkatan 100
\(10^{300} = (10^3)^{100} = 1000^{100}\)
\(21^{200} = (21^2)^{100} = 441^{100}\)
\(35^{100}\)
\(2^{500} = (2^5)^{100} = 10^{100}\)
Urutan bilangan dari yang terkecil sampai terbesar dapat kita lihat dari urutan bilangan-bilangan pokoknya
\(10^{100};\:\:\:\:35^{100};\:\:\:441^{100};\:\:\:1000^{100}\)
atau sama dengan
\(2^{500};\:\:\:\:35^{100};\:\:\:21^{200};\:\:\:10^{300}\)
Jawaban: C
\(\dfrac{1}{\dfrac{1}{10^5} + 1} + \dfrac{1}{\dfrac{1}{10^4} + 1} + \dotso + \dfrac{1}{10^{4} + 1} + \dfrac{1}{10^{5} + 1}\)
\(\dfrac{1}{\dfrac{1}{10^5} + 1} + \dfrac{1}{\dfrac{1}{10^4} + 1} + \dotso + \dfrac{1}{10^{4} + 1} + \dfrac{1}{10^{5} + 1}\)
\(\dfrac{1}{\dfrac{1}{10^5} + \dfrac{10^5}{10^5}} + \dfrac{1}{\dfrac{1}{10^4} + \dfrac{10^4}{10^4}} + \dotso + \dfrac{1}{10^{4} + 1} + \dfrac{1}{10^{5} + 1}\)
\(\dfrac{1}{\dfrac{1 + 10^5}{10^5}} + \dfrac{1}{\dfrac{1 + 10^4}{10^4}} + \dotso + \dfrac{1}{10^{4} + 1} + \dfrac{1}{10^{5} + 1}\)
\(\color{blue}\dfrac{10^5}{1 + 10^5} \color{purple}+ \dfrac{10^4}{1 + 10^4}\color{black} + \dotso + \color{purple}\dfrac{1}{10^{4} + 1} + \color{blue}\dfrac{1}{10^{5} + 1}\)
Perhatikan pola penjumlahan suku depan dan suku belakang:
\(\color{blue} \dfrac{10^5}{1 + 10^5} + \dfrac{1}{10^{5} + 1} = \dfrac{10^5 + 1}{1 + 10^5} = 1\)
\(\color{purple}\dfrac{10^4}{1 + 10^4} + \dfrac{1}{10^{4} + 1} = \dfrac{10^4 + 1}{1 + 10^4} = 1\)
demikian seterusnya, dan ada satu suku yang tidak memiliki pasangan, yaitu \(\dfrac{1}{10^0 + 1}\)
\(1 + 1 + 1 + 1 + 1 + \dfrac{1}{10^0 + 1}\)
\(5 + \dfrac{1}{1 + 1}\)
\(5 + \dfrac{1}{2}\)
\(5\dfrac{1}{2}\)
