Soalan 1:
Hitung nilai setiap yang berikut tanpa menggunakan kalkulator.
$$ \text { (a) } 4^{\frac{1}{3}} \times 50^{\frac{2}{3}} \times 10^{\frac{5}{3}} $$
$$ \text { (b) } 5^{\frac{5}{2}} \times 20^{\frac{3}{2}} \div 10^{-2} $$
$$ \text { (c) } 60^{\frac{1}{2}} \times 125^{\frac{2}{3}} \div \sqrt{15} $$
Penyelesaian:
(a)
$$ \begin{aligned} & 4^{\frac{1}{3}} \times 50^{\frac{2}{3}} \times 10^{\frac{5}{3}} \\ & =\left(2^2\right)^{\frac{1}{3}} \times\left(2^1 \times 5^2\right)^{\frac{2}{3}} \times\left(2^1 \times 5^1\right)^{\frac{5}{3}} \\ & =2^{\frac{2}{3}} \times 2^{\frac{2}{3}} \times 5^{\frac{4}{3}} \times 2^{\frac{5}{3}} \times 5^{\frac{5}{3}} \end{aligned} $$
$$ \begin{aligned} & =2^{\frac{2}{3}+\frac{2}{3}+\frac{5}{3}} \times 5^{\frac{4}{3}+\frac{5}{3}} \\ & =2^{\frac{9}{3}} \times 5^{\frac{9}{3}} \\ & =2^3 \times 5^3 \\ & =(2 \times 5)^3 \\ & =(10)^3 \\ & =1000 \end{aligned} $$
(b)
$$ \begin{aligned} & 5^{\frac{5}{2}} \times 20^{\frac{3}{2}} \div 10^{-2} \\ & =5^{\frac{5}{2}} \times\left(2^2 \times 5^1\right)^{\frac{3}{2}} \div\left(2^1 \times 5^1\right)^{-2} \\ & =5^{\frac{5}{2}} \times 2^3 \times 5^{\frac{3}{2}} \div 2^{-2} \div 5^{-2} \\ & =5^{\frac{5}{2}+\frac{3}{2}-(-2)} \times 2^{3-(-2)} \\ & =5^{\frac{8}{2}+2} \times 2^{3+2} \\ & =5^6 2^5 \\ & =5^5 \times 5^1 \times 2^5 \\ & =(5 \times 2)^5 \times 5^1 \\ & =(10)^5 \times 5^1 \\ & =100000 \times 5 \\ & =500000 \end{aligned} $$
(c)
$$ \begin{aligned} & 60^{\frac{1}{2}} \times 125^{\frac{2}{3}} \div \sqrt{15} \\ & =\left(2^2 \times 3 \times 5\right)^{\frac{1}{2}} \times\left(5^3\right)^{\frac{2}{3}} \div(3 \times 5)^{\frac{1}{2}} \\ & =2^{\frac{2}{2}} \times 3^{\frac{1}{2}} \times 5^{\frac{1}{2}} \times 5^2 \div 3^{\frac{1}{2}} \div 5^{\frac{1}{2}} \\ & =2^1 \times 3^{\frac{1}{2}-\frac{1}{2}} \times 5^{\frac{1}{2}+2-\frac{1}{2}} \\ & =2 \times 3^0 \times 5^2 \\ & =2 \times 1 \times 25 \\ & =50 \end{aligned} $$
Hitung nilai setiap yang berikut tanpa menggunakan kalkulator.
$$ \text { (a) } 4^{\frac{1}{3}} \times 50^{\frac{2}{3}} \times 10^{\frac{5}{3}} $$
$$ \text { (b) } 5^{\frac{5}{2}} \times 20^{\frac{3}{2}} \div 10^{-2} $$
$$ \text { (c) } 60^{\frac{1}{2}} \times 125^{\frac{2}{3}} \div \sqrt{15} $$
Penyelesaian:
(a)
$$ \begin{aligned} & 4^{\frac{1}{3}} \times 50^{\frac{2}{3}} \times 10^{\frac{5}{3}} \\ & =\left(2^2\right)^{\frac{1}{3}} \times\left(2^1 \times 5^2\right)^{\frac{2}{3}} \times\left(2^1 \times 5^1\right)^{\frac{5}{3}} \\ & =2^{\frac{2}{3}} \times 2^{\frac{2}{3}} \times 5^{\frac{4}{3}} \times 2^{\frac{5}{3}} \times 5^{\frac{5}{3}} \end{aligned} $$
$$ \begin{aligned} & =2^{\frac{2}{3}+\frac{2}{3}+\frac{5}{3}} \times 5^{\frac{4}{3}+\frac{5}{3}} \\ & =2^{\frac{9}{3}} \times 5^{\frac{9}{3}} \\ & =2^3 \times 5^3 \\ & =(2 \times 5)^3 \\ & =(10)^3 \\ & =1000 \end{aligned} $$
(b)
$$ \begin{aligned} & 5^{\frac{5}{2}} \times 20^{\frac{3}{2}} \div 10^{-2} \\ & =5^{\frac{5}{2}} \times\left(2^2 \times 5^1\right)^{\frac{3}{2}} \div\left(2^1 \times 5^1\right)^{-2} \\ & =5^{\frac{5}{2}} \times 2^3 \times 5^{\frac{3}{2}} \div 2^{-2} \div 5^{-2} \\ & =5^{\frac{5}{2}+\frac{3}{2}-(-2)} \times 2^{3-(-2)} \\ & =5^{\frac{8}{2}+2} \times 2^{3+2} \\ & =5^6 2^5 \\ & =5^5 \times 5^1 \times 2^5 \\ & =(5 \times 2)^5 \times 5^1 \\ & =(10)^5 \times 5^1 \\ & =100000 \times 5 \\ & =500000 \end{aligned} $$
(c)
$$ \begin{aligned} & 60^{\frac{1}{2}} \times 125^{\frac{2}{3}} \div \sqrt{15} \\ & =\left(2^2 \times 3 \times 5\right)^{\frac{1}{2}} \times\left(5^3\right)^{\frac{2}{3}} \div(3 \times 5)^{\frac{1}{2}} \\ & =2^{\frac{2}{2}} \times 3^{\frac{1}{2}} \times 5^{\frac{1}{2}} \times 5^2 \div 3^{\frac{1}{2}} \div 5^{\frac{1}{2}} \\ & =2^1 \times 3^{\frac{1}{2}-\frac{1}{2}} \times 5^{\frac{1}{2}+2-\frac{1}{2}} \\ & =2 \times 3^0 \times 5^2 \\ & =2 \times 1 \times 25 \\ & =50 \end{aligned} $$
Soalan 2:
Hitung nilai x bagi setiap persamaan berikut.
$$ \text { (a) } 64 x^{\frac{1}{2}}=27 x^{-\frac{5}{2}} $$
$$ \text { (b) } 3 x^{\frac{2}{3}}=\frac{27}{4} x^{-\frac{4}{3}} $$
$$ \text { (c) } 25 x^{-\frac{2}{3}}-\frac{5}{3} x^{\frac{1}{3}}=0 $$
Penyelesaian:
(a)
$$ \begin{aligned} 64 x^{\frac{1}{2}} & =27 x^{-\frac{5}{2}} \\ 4^3 x^{\frac{1}{2}} & =3^3 x^{-\frac{5}{2}} \\ \frac{x^{\frac{1}{2}}}{x^{-\frac{5}{2}}} & =\frac{3^3}{4^3} \\ x^{\frac{1}{2}-\left(-\frac{5}{2}\right)} & =\left(\frac{3}{4}\right)^3 \end{aligned} $$
$$ \begin{aligned} x^{\frac{6}{2}} & =\left(\frac{3}{4}\right)^3 \\ x^3 & =\left(\frac{3}{4}\right)^3 \\ x & =\left(\frac{3}{4}\right)^{3 \times \frac{1}{3}} \\ x & =\frac{3}{4} \end{aligned} $$
(b)
$$ \begin{aligned} 3 x^{\frac{2}{3}} & =\frac{27}{4} x^{-\frac{4}{3}} \\ (\div 3), x^{\frac{2}{3}} & =\frac{9}{4} x^{-\frac{4}{3}} \\ x^{\frac{2}{3}} & =\left(\frac{3}{2}\right)^2 x^{-\frac{4}{3}} \\ \frac{x^{\frac{2}{3}}}{x^{-\frac{4}{3}}} & =\left(\frac{3}{2}\right)^2 \end{aligned} $$
$$ \begin{aligned} x^{\frac{2}{3}-\left(-\frac{4}{3}\right)} & =\left(\frac{3}{2}\right)^2 \\ x^{\frac{2}{3}+\frac{4}{3}} & =\left(\frac{3}{2}\right)^2 \\ x^2 & =\left(\frac{3}{2}\right)^2 \\ x & =\left(\frac{3}{2}\right)^{2 \times \frac{1}{2}} \\ x & =\frac{3}{2} \end{aligned} $$
(c)
$$ \begin{aligned} & 25 x^{-\frac{2}{3}}-\frac{5}{3} x^{\frac{1}{3}}=0 \\ & 25 x^{-\frac{2}{3}}=\frac{5}{3} x^{\frac{1}{3}} \\ & 25 \times \frac{3}{5}=\frac{x^{\frac{1}{3}}}{x^{-\frac{2}{3}}} \\ & 15=x^{\frac{1}{3}-\left(-\frac{2}{3}\right)} \\ & 15=x^{\frac{1}{3}+\frac{2}{3}} \\ & 15=x^1 \\ & x=15 \end{aligned} $$
Hitung nilai x bagi setiap persamaan berikut.
$$ \text { (a) } 64 x^{\frac{1}{2}}=27 x^{-\frac{5}{2}} $$
$$ \text { (b) } 3 x^{\frac{2}{3}}=\frac{27}{4} x^{-\frac{4}{3}} $$
$$ \text { (c) } 25 x^{-\frac{2}{3}}-\frac{5}{3} x^{\frac{1}{3}}=0 $$
Penyelesaian:
(a)
$$ \begin{aligned} 64 x^{\frac{1}{2}} & =27 x^{-\frac{5}{2}} \\ 4^3 x^{\frac{1}{2}} & =3^3 x^{-\frac{5}{2}} \\ \frac{x^{\frac{1}{2}}}{x^{-\frac{5}{2}}} & =\frac{3^3}{4^3} \\ x^{\frac{1}{2}-\left(-\frac{5}{2}\right)} & =\left(\frac{3}{4}\right)^3 \end{aligned} $$
$$ \begin{aligned} x^{\frac{6}{2}} & =\left(\frac{3}{4}\right)^3 \\ x^3 & =\left(\frac{3}{4}\right)^3 \\ x & =\left(\frac{3}{4}\right)^{3 \times \frac{1}{3}} \\ x & =\frac{3}{4} \end{aligned} $$
(b)
$$ \begin{aligned} 3 x^{\frac{2}{3}} & =\frac{27}{4} x^{-\frac{4}{3}} \\ (\div 3), x^{\frac{2}{3}} & =\frac{9}{4} x^{-\frac{4}{3}} \\ x^{\frac{2}{3}} & =\left(\frac{3}{2}\right)^2 x^{-\frac{4}{3}} \\ \frac{x^{\frac{2}{3}}}{x^{-\frac{4}{3}}} & =\left(\frac{3}{2}\right)^2 \end{aligned} $$
$$ \begin{aligned} x^{\frac{2}{3}-\left(-\frac{4}{3}\right)} & =\left(\frac{3}{2}\right)^2 \\ x^{\frac{2}{3}+\frac{4}{3}} & =\left(\frac{3}{2}\right)^2 \\ x^2 & =\left(\frac{3}{2}\right)^2 \\ x & =\left(\frac{3}{2}\right)^{2 \times \frac{1}{2}} \\ x & =\frac{3}{2} \end{aligned} $$
(c)
$$ \begin{aligned} & 25 x^{-\frac{2}{3}}-\frac{5}{3} x^{\frac{1}{3}}=0 \\ & 25 x^{-\frac{2}{3}}=\frac{5}{3} x^{\frac{1}{3}} \\ & 25 \times \frac{3}{5}=\frac{x^{\frac{1}{3}}}{x^{-\frac{2}{3}}} \\ & 15=x^{\frac{1}{3}-\left(-\frac{2}{3}\right)} \\ & 15=x^{\frac{1}{3}+\frac{2}{3}} \\ & 15=x^1 \\ & x=15 \end{aligned} $$