Contoh Soal Sifat-sifat Eksponen dan Pembahasannya #2

Hai sob, pada postingan kali ini, mimin sajikan lanjutan beberapa contoh soal dan pembahasan pada materi sifat-sifat eksponen. Cuss, langsung saja. Berikut contoh-contoh soal dan pembahasannya. Selamat belajar. Semoga bermanfaat.

Untuk pengingat, sifat-sifat eksponen diantaranya adalah sebagai berikut:

Contoh soal 1

Tentukan nilai dari:

$\begin{array}{rl}& a{.16}^{\frac{1}{2}}+{27}^{\frac{1}{3}}-{49}^{\frac{1}{2}}\\ & b.{81}^{\frac{1}{4}}-{125}^{\frac{1}{3}}+4^{\frac{3}{2}}\\ & c.{27}^{\frac{2}{3}}-{16}^{\frac{1}{2}}+{25}^{\frac{3}{2}}\\ & d.(\frac{1}{4}{)}^{-\frac{3}{2}}+{16}^{\frac{5}{4}}+(\frac{1}{8}{)}^{-\frac{1}{3}}\\ & e.(\frac{4}{9}{)}^{\frac{1}{2}}+(\frac{8}{27}{)}^{\frac{1}{3}}+(\frac{16}{81}{)}^{\frac{1}{4}}\\ & f.(\frac{1}{4}{)}^{-\frac{1}{2}}+(\frac{1}{27}{)}^{-\frac{1}{3}}+(\frac{1}{256}{)}^{-\frac{1}{4}}\\ & g{.8}^3\times 4^5:2^7\end{array}$

Jawab:

$\begin{array}{rl}& a.{16}^{\frac{1}{2}}+{27}^{\frac{1}{3}}-{49}^{\frac{1}{2}}\\ & =4^{2\times \frac{1}{2}}+3^{3\times \frac{1}{3}}-7^{2\times \frac{1}{2}}\\ & =4+3-7\\ & =0\\ \\ & b.{81}^{\frac{1}{4}}-{125}^{\frac{1}{3}}+4^{\frac{3}{2}}\\ & =3^{4\times \frac{1}{4}}-5^{3\times \frac{1}{3}}+2^{2\times \frac{3}{2}}\\ & =3-5+8\\ & =6\\ \\ & c.{27}^{\frac{2}{3}}-{16}^{\frac{1}{2}}+{25}^{\frac{3}{2}}\\ & =3^{3\times \frac{2}{3}}-4^{2\times \frac{1}{2}}+5^{2\times \frac{3}{2}}\\ & =9-4+125\\ & =130\\ \\ & d.(\frac{1}{4}{)}^{-\frac{3}{2}}+{16}^{\frac{5}{4}}+(\frac{1}{8}{)}^{-\frac{1}{3}}\\ & =2^{-2\times -\frac{3}{2}}+2^{4\times \frac{5}{4}}+2^{-3\times -\frac{1}{3}}\\ & =8+32+2\\ & =42\\ \\ & e.(\frac{4}{9}{)}^{\frac{1}{2}}+(\frac{8}{27}{)}^{\frac{1}{3}}+(\frac{16}{81}{)}^{\frac{1}{4}}\\ & =(\frac{2}{3}{)}^{2\times \frac{1}{2}}+(\frac{2}{3}{)}^{3\times \frac{1}{3}}+(\frac{2}{3}{)}^{4\times \frac{1}{4}}\\ & =\frac{2}{3}+\frac{2}{3}+\frac{2}{3}\\ & =\frac{6}{3}\\ & =2\\ \\ & f.(\frac{1}{4}{)}^{-\frac{1}{2}}+(\frac{1}{27}{)}^{-\frac{1}{3}}+(\frac{1}{256}{)}^{-\frac{1}{4}}\\ & =(2{)}^{-2\times -\frac{1}{2}}+3^{-3\times -\frac{1}{3}}+4^{-4\times -\frac{1}{4}}\\ & =2+3+4\\ & =9\\ \\ & g.8^3\times 4^5:2^7\\ & =(2^3{)}^3\times (2^2{)}^5:2^7\\ & =2^9\times 2^{10}:2^7\\ & =2^{9+10-2}\\ & =2^{17}\end{array}$

Contoh soal 2

Tentukan hasil dari $\frac{x^5{y}^{-1}{z}^{-4}}{(xyz{)}^{-6}}$ 

Jawab:

$\begin{array}{rl}\frac{x^5{y}^{-1}{z}^{-4}}{(xyz{)}^{-6}}& =\frac{x^5{y}^{-1}{z}^{-4}}{x^{-6}{y}^{-6}{z}^{-6}}\\ & =x^{5+6}{y}^{-1+6}{z}^{-4+6}\\ & =x^{11}{y}^5{z}^2\end{array}$

Contoh soal 3 

Tentukan bentuk sederhana dari:

$\begin{array}{rl}& a.3^5\times 3^4\times 3^0\\ & b.\frac{2^7\times 2^3}{2^5}\\ & c.(\frac{5^{-2}}{6}{)}^3\times (5^4\times 6{)}^3\\ & d.\frac{4^3\times 2^{-6}}{8^{-2}\times 64}\\ & e.2^{\frac{2}{3}}\times 4^2\\ & f.\frac{\sqrt[4]{2}\times \sqrt[2]{2^5}}{2}\\ & g.(\frac{a^4\times a^{-2}}{a^3}{)}^2\\ & h.\frac{(x^{-2}{y}^{3}z{)}^2}{x{y}^{-2}{z}^3}\\ & i.\frac{4-(2b^2{)}^0}{2}\\ & j.(\frac{(3m\times 3m^2)}{3^2\times m^2}{)}^{-2}\end{array}$

Jawab:

$\begin{array}{rl}& a.3^5\times 3^4\times 3^0\\ & =3^{(5+4+0)}\\ & =3^9\\ \\ & b.\frac{2^7\times 2^3}{2^5}\\ & =2^7\times 2^3\times 2^{-5}\\ & =2^{(7+3-5)}\\ & =2^5\\ \\ & c.(\frac{5^{-2}}{6}{)}^3\times (5^4\times 6{)}^3\\ & =(5^{-2}\times 6^{-1}{)}^3\times (5^4\times 6{)}^3\\ & =5^{-6}\times 6^{-3}\times 5^{12}\times 6^3\\ & =5^{(-6+12)}\times 6^{(-3+3)}\\ & =5^6\times 5^0\\ & =5^6\\ \\ & d.\frac{4^3\times 2^{-6}}{8^{-2}\times 64}\\ & =\frac{2^6\times 2^{-6}}{2^{-6}\times 2^4}\\ & =\frac{2^{(6-6)}}{2^{(-6+4)}}\\ & =\frac{2^0}{2^{-2}}\\ & =4\\ \\ & e.2^{\frac{2}{3}}\times 4^2\\ & =2^{\frac{2}{3}}\times 2^4\\ & =2^{(\frac{2}{3}+4)}\\ & =2^{\frac{14}{3}}\\ \\ & f.\frac{\sqrt[4]{2}\times \sqrt[2]{2^5}}{2}\\ & =2^{\frac{1}{4}}\times 2^{\frac{5}{2}}\times 2^{-1}\\ & =2^{\frac{7}{4}}\\ \\ & g.(\frac{a^4\times a^{-2}}{a^3}{)}^2\\ & =\frac{a^8\times a^{-4}}{a^6}\\ & =a^8\times a^{-4}\times a^{-6}\\ & =a^{(8-4-6)}\\ & =a^{-2}\\ & =\frac{1}{a^2}\\ \\ & h.\frac{(x^{-2}{y}^{3}z{)}^2}{x{y}^{-2}{z}^3}\\ & =\frac{x^{-4}{y}^6{z}^2}{x{y}^{-2}{z}^3}\\ & =x^{(-4-1)}{y}^{(6+2)}{z}^{(2-3)}\\ & =x^{-5}{y}^8{z}^{-1}\\ & =\frac{y^8}{x^{5}z}\\ \\ & i.\frac{4-(2b^2{)}^0}{2}\\ & =\frac{4-1}{2}\\ & =\frac{3}{2}\\ \\ & j.(\frac{(3m\times 3m^2)}{3^2\times m^2}{)}^{-2}\\ & =\frac{(3m\times 3m^2{)}^{-4}}{(3^2\times m^2{)}^{-2}}\\ & =\frac{(3^2\times m^3{)}^{-4}}{(3^2\times m^2{)}^{-2}}\\ & =\frac{3^{-8}\times m^{-12}}{3^{-4}\times m^{-4}}\\ & =3^{-8+4}\times m^{-12+4}\\ & =3^4{m}^{-8}\\ & =\frac{81}{m^8}\end{array}$

Demikianlah beberapa contoh soal dan pembahasan pada materi sifat-sifat eksponen. Semoga bermanfaat.