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:

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

Jawab:

$ \begin{aligned} &a. 16^{\frac{1}{2}}+27^{\frac{1}{3}}-49^{\frac{1}{2}}\\&= \color{red}{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}}\\&= \color{red}{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}}\\&= \color{red}{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}}\\&= \color{red}{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}}\\&= \color{red}{(\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}}\\&= \color{red}{(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 \\ &= \color{red}{(2^3)^3 \times (2^2)^5 : 2^7} \\&= \color{red}{2^9 \times 2^{10} : 2^7}  \\&= 2^{9+10-2}\\&=2^{17} \end{aligned}$

Contoh soal 2

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

Jawab:

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

Contoh soal 3 

Tentukan bentuk sederhana dari:

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

Jawab:

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

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