Soalan 2:
Hitung nilai setiap yang berikut.
$$ \text { (a) } \frac{\sqrt{7^{-4} \times 11^4}}{49 \times 121} $$
$$ \text { (b) } \frac{\left(5^{-3} \times 3^6\right)^{\frac{1}{3}} \times \sqrt[4]{16}}{(125 \times 729 \times 64)^{-\frac{1}{3}}} $$
$$ \text { (c) } \frac{\left(2^6 \times 3^4 \times 5^2\right)^{\frac{3}{2}}}{\sqrt[4]{256} \times \sqrt{729} \times \sqrt[3]{125}} $$
$$ \text { (d) } \frac{9 \sqrt{512} \times \sqrt[3]{343} \times \sqrt{121}}{(64)^{\frac{1}{3}} \times(81)^{\frac{3}{4}} \times(14641)^{\frac{1}{4}}} $$
$$ \text { (e) } \frac{\left(2^4 \times 3^6\right)^{\frac{1}{2}} \times 3 \sqrt{8} \times \sqrt{81}}{16^{\frac{3}{4}} \times 27^{\frac{1}{3}}} $$
$$ \text { (f) } \frac{64^{\frac{2}{3}} \times \sqrt[3]{125} \times\left(2 \times \frac{1}{5}\right)^{-3}}{4^2 \times \sqrt[4]{625}} $$
Penyelesaian:
(a)
$$ \begin{aligned} & \frac{\sqrt{7^{-4} \times 11^4}}{49 \times 121} \\ & =\frac{\left(7^{-4} \times 11^4\right)^{\frac{1}{2}}}{49 \times 121} \\ & =\frac{7^{-4 \times \frac{1}{2}} \times 11^{4 \times \frac{1}{2}}}{7^2 \times 11^2} \end{aligned} $$
$$ \begin{aligned} & =\frac{7^{-2} \times 11^2}{7^2 \times 11^2} \\ & =7^{-2-2} \times 11^{2-2} \\ & =7^{-4} \times 11^0 \\ & =\frac{1}{7^4} \times 1 \\ & =\frac{1}{2401} \end{aligned} $$
(b)
$$ \begin{aligned} & \frac{\left(5^{-3} \times 3^6\right)^{\frac{1}{3}} \times \sqrt[4]{16}}{(125 \times 729 \times 64)^{-\frac{1}{3}}} \\ & =\frac{5^{-\frac{3}{3}} \times 3^{\frac{6}{3}} \times 16^{\frac{1}{4}}}{125^{-\frac{1}{3}} \times 729^{-\frac{1}{3}} \times 64^{-\frac{1}{3}}} \\ & =5^{-1} \times 3^2 \times 4^{2 \times \frac{1}{4}} \times 125^{\frac{1}{3}} \times 729^{\frac{1}{3}} \times 64^{\frac{1}{3}} \end{aligned} $$
$$ \begin{aligned} & =5^{-1} \times 3^2 \times 4^{\frac{1}{2}} \times 5^{3 \times \frac{1}{3}} \times 3^{6 \times \frac{1}{3}} \times 4^{3 \times \frac{1}{3}} \\ & =5^{-1} \times 3^2 \times 2 \times 5^1 \times 3^2 \times 4^1 \\ & =5^{-1+1} \times 3^{2+2} \times 2 \times 4 \\ & =5^0 \times 3^4 \times 8 \\ & =1 \times 81 \times 8 \\ & =648 \end{aligned} $$
(c)
$$ \begin{aligned} & \frac{\left(2^6 \times 3^4 \times 5^2\right)^{\frac{3}{2}}}{\sqrt[4]{256} \times \sqrt{729} \times \sqrt[3]{125}} \\ = & \frac{2^{6 \times \frac{3}{2}} \times 3^{4 \times \frac{3}{2}} \times 5^{2 \times \frac{3}{2}}}{256^{\frac{1}{4}} \times 729^{\frac{1}{2}} \times 125^{\frac{1}{3}}} \\ = & \frac{2^9 \times 3^6 \times 5^3}{4^{4 \times \frac{1}{4}} \times 27^{2 \times \frac{1}{2}} \times 5^{3 \times \frac{1}{3}}} \end{aligned} $$
$$ \begin{aligned} & =\frac{2^9 \times 3^6 \times 5^3}{4 \times 27 \times 5} \\ & =\frac{2^9 \times 3^6 \times 5^3}{2^2 \times 3^3 \times 5^1} \\ & =2^{9-2} \times 3^{6-3} \times 5^{3-1} \\ & =2^7 \times 3^3 \times 5^2 \\ & =86400 \end{aligned} $$
(d)
$$ \begin{aligned} & \frac{\sqrt[9]{512} \times \sqrt[3]{343} \times \sqrt{121}}{(64)^{\frac{1}{3}} \times(81)^{\frac{3}{4}} \times(14641)^{\frac{1}{4}}} \\ & =\frac{(512)^{\frac{1}{9}} \times(343)^{\frac{1}{3}} \times(121)^{\frac{1}{2}}}{(64)^{\frac{1}{3}} \times(81)^{\frac{3}{4}} \times(14641)^{\frac{1}{4}}} \end{aligned} $$
$$ \begin{aligned} & =\frac{2^{9 \times \frac{1}{9}} \times 7^{3 \times \frac{1}{3}} \times 11^{2 \times \frac{1}{2}}}{4^{3 \times \frac{1}{3}} \times 3^{4 \times \frac{3}{4}} \times 11^{4 \times \frac{1}{4}}} \\ & =\frac{2 \times 7 \times 11}{4 \times 3^3 \times 11} \\ & =\frac{7}{2 \times 3^3} \\ & =\frac{7}{54} \end{aligned} $$
(e)
$$ \begin{aligned} & \frac{\left(2^4 \times 3^6\right)^{\frac{1}{2}} \times \sqrt[3]{8} \times \sqrt{81}}{16^{\frac{3}{4}} \times 27^{\frac{1}{3}}} \\ & =\frac{2^{4 \times \frac{1}{2}} \times 3^{6 \times \frac{1}{2}} \times 8^{\frac{1}{3}} \times 81^{\frac{1}{2}}}{16^{\frac{1}{4} \times 3} \times 27^{\frac{1}{3}}} \end{aligned} $$
$$ \begin{aligned} & =\frac{2^2 \times 3^3 \times 2 \times 9}{2^3 \times 3} \\ & =\frac{2^2 \times 3^3 \times 2 \times 3^2}{2^3 \times 3} \\ & =2^{2+1-3} \times 3^{3+2-1} \\ & =2^0 \times 3^4 \\ & =81 \end{aligned} $$
(f)
$$ \begin{aligned} & \frac{64^{\frac{2}{3}} \times \sqrt[3]{125} \times\left(2 \times \frac{1}{5}\right)^{-3}}{4^2 \times \sqrt[4]{625}} \\ & =\frac{\left(4^3\right)^{\frac{2}{3}} \times 125^{\frac{1}{3}} \times 2^{-3} \times\left(\frac{1}{5}\right)^{-3}}{4^2 \times 625^{\frac{1}{4}}} \\ & =\frac{4^2 \times 5 \times 2^{-3} \times 5^3}{4^2 \times 5^{4 \times \frac{1}{4}}} \end{aligned} $$
$$ \begin{aligned} & =\frac{2^4 \times 5 \times 2^{-3} \times 5^3}{2^4 \times 5} \\ & =2^{4+(-3)-4} \times 5^{1+3-1} \\ & =2^{-3} \times 5^3 \\ & =\frac{5^3}{2^3} \\ & =\frac{125}{8} \end{aligned} $$
Hitung nilai setiap yang berikut.
$$ \text { (a) } \frac{\sqrt{7^{-4} \times 11^4}}{49 \times 121} $$
$$ \text { (b) } \frac{\left(5^{-3} \times 3^6\right)^{\frac{1}{3}} \times \sqrt[4]{16}}{(125 \times 729 \times 64)^{-\frac{1}{3}}} $$
$$ \text { (c) } \frac{\left(2^6 \times 3^4 \times 5^2\right)^{\frac{3}{2}}}{\sqrt[4]{256} \times \sqrt{729} \times \sqrt[3]{125}} $$
$$ \text { (d) } \frac{9 \sqrt{512} \times \sqrt[3]{343} \times \sqrt{121}}{(64)^{\frac{1}{3}} \times(81)^{\frac{3}{4}} \times(14641)^{\frac{1}{4}}} $$
$$ \text { (e) } \frac{\left(2^4 \times 3^6\right)^{\frac{1}{2}} \times 3 \sqrt{8} \times \sqrt{81}}{16^{\frac{3}{4}} \times 27^{\frac{1}{3}}} $$
$$ \text { (f) } \frac{64^{\frac{2}{3}} \times \sqrt[3]{125} \times\left(2 \times \frac{1}{5}\right)^{-3}}{4^2 \times \sqrt[4]{625}} $$
Penyelesaian:
(a)
$$ \begin{aligned} & \frac{\sqrt{7^{-4} \times 11^4}}{49 \times 121} \\ & =\frac{\left(7^{-4} \times 11^4\right)^{\frac{1}{2}}}{49 \times 121} \\ & =\frac{7^{-4 \times \frac{1}{2}} \times 11^{4 \times \frac{1}{2}}}{7^2 \times 11^2} \end{aligned} $$
$$ \begin{aligned} & =\frac{7^{-2} \times 11^2}{7^2 \times 11^2} \\ & =7^{-2-2} \times 11^{2-2} \\ & =7^{-4} \times 11^0 \\ & =\frac{1}{7^4} \times 1 \\ & =\frac{1}{2401} \end{aligned} $$
(b)
$$ \begin{aligned} & \frac{\left(5^{-3} \times 3^6\right)^{\frac{1}{3}} \times \sqrt[4]{16}}{(125 \times 729 \times 64)^{-\frac{1}{3}}} \\ & =\frac{5^{-\frac{3}{3}} \times 3^{\frac{6}{3}} \times 16^{\frac{1}{4}}}{125^{-\frac{1}{3}} \times 729^{-\frac{1}{3}} \times 64^{-\frac{1}{3}}} \\ & =5^{-1} \times 3^2 \times 4^{2 \times \frac{1}{4}} \times 125^{\frac{1}{3}} \times 729^{\frac{1}{3}} \times 64^{\frac{1}{3}} \end{aligned} $$
$$ \begin{aligned} & =5^{-1} \times 3^2 \times 4^{\frac{1}{2}} \times 5^{3 \times \frac{1}{3}} \times 3^{6 \times \frac{1}{3}} \times 4^{3 \times \frac{1}{3}} \\ & =5^{-1} \times 3^2 \times 2 \times 5^1 \times 3^2 \times 4^1 \\ & =5^{-1+1} \times 3^{2+2} \times 2 \times 4 \\ & =5^0 \times 3^4 \times 8 \\ & =1 \times 81 \times 8 \\ & =648 \end{aligned} $$
(c)
$$ \begin{aligned} & \frac{\left(2^6 \times 3^4 \times 5^2\right)^{\frac{3}{2}}}{\sqrt[4]{256} \times \sqrt{729} \times \sqrt[3]{125}} \\ = & \frac{2^{6 \times \frac{3}{2}} \times 3^{4 \times \frac{3}{2}} \times 5^{2 \times \frac{3}{2}}}{256^{\frac{1}{4}} \times 729^{\frac{1}{2}} \times 125^{\frac{1}{3}}} \\ = & \frac{2^9 \times 3^6 \times 5^3}{4^{4 \times \frac{1}{4}} \times 27^{2 \times \frac{1}{2}} \times 5^{3 \times \frac{1}{3}}} \end{aligned} $$
$$ \begin{aligned} & =\frac{2^9 \times 3^6 \times 5^3}{4 \times 27 \times 5} \\ & =\frac{2^9 \times 3^6 \times 5^3}{2^2 \times 3^3 \times 5^1} \\ & =2^{9-2} \times 3^{6-3} \times 5^{3-1} \\ & =2^7 \times 3^3 \times 5^2 \\ & =86400 \end{aligned} $$
(d)
$$ \begin{aligned} & \frac{\sqrt[9]{512} \times \sqrt[3]{343} \times \sqrt{121}}{(64)^{\frac{1}{3}} \times(81)^{\frac{3}{4}} \times(14641)^{\frac{1}{4}}} \\ & =\frac{(512)^{\frac{1}{9}} \times(343)^{\frac{1}{3}} \times(121)^{\frac{1}{2}}}{(64)^{\frac{1}{3}} \times(81)^{\frac{3}{4}} \times(14641)^{\frac{1}{4}}} \end{aligned} $$
$$ \begin{aligned} & =\frac{2^{9 \times \frac{1}{9}} \times 7^{3 \times \frac{1}{3}} \times 11^{2 \times \frac{1}{2}}}{4^{3 \times \frac{1}{3}} \times 3^{4 \times \frac{3}{4}} \times 11^{4 \times \frac{1}{4}}} \\ & =\frac{2 \times 7 \times 11}{4 \times 3^3 \times 11} \\ & =\frac{7}{2 \times 3^3} \\ & =\frac{7}{54} \end{aligned} $$
(e)
$$ \begin{aligned} & \frac{\left(2^4 \times 3^6\right)^{\frac{1}{2}} \times \sqrt[3]{8} \times \sqrt{81}}{16^{\frac{3}{4}} \times 27^{\frac{1}{3}}} \\ & =\frac{2^{4 \times \frac{1}{2}} \times 3^{6 \times \frac{1}{2}} \times 8^{\frac{1}{3}} \times 81^{\frac{1}{2}}}{16^{\frac{1}{4} \times 3} \times 27^{\frac{1}{3}}} \end{aligned} $$
$$ \begin{aligned} & =\frac{2^2 \times 3^3 \times 2 \times 9}{2^3 \times 3} \\ & =\frac{2^2 \times 3^3 \times 2 \times 3^2}{2^3 \times 3} \\ & =2^{2+1-3} \times 3^{3+2-1} \\ & =2^0 \times 3^4 \\ & =81 \end{aligned} $$
(f)
$$ \begin{aligned} & \frac{64^{\frac{2}{3}} \times \sqrt[3]{125} \times\left(2 \times \frac{1}{5}\right)^{-3}}{4^2 \times \sqrt[4]{625}} \\ & =\frac{\left(4^3\right)^{\frac{2}{3}} \times 125^{\frac{1}{3}} \times 2^{-3} \times\left(\frac{1}{5}\right)^{-3}}{4^2 \times 625^{\frac{1}{4}}} \\ & =\frac{4^2 \times 5 \times 2^{-3} \times 5^3}{4^2 \times 5^{4 \times \frac{1}{4}}} \end{aligned} $$
$$ \begin{aligned} & =\frac{2^4 \times 5 \times 2^{-3} \times 5^3}{2^4 \times 5} \\ & =2^{4+(-3)-4} \times 5^{1+3-1} \\ & =2^{-3} \times 5^3 \\ & =\frac{5^3}{2^3} \\ & =\frac{125}{8} \end{aligned} $$