Soalan 1:
Ringkaskan setiap yang berikut.
Penyelesaian:
(a)
$$ \begin{aligned} & \left(2 \times 3^4\right)^2 \\ & =2^{1(2)} \times 3^{4(2)} \\ & =2^2 \times 3^8 \end{aligned} $$
(b)
$$ \begin{aligned} & \left(11^3 \times 9^5\right)^3 \\ & =11^{3(3)} \times 9^{5(3)} \\ & =11^9 \times 9^{15} \end{aligned} $$
(c)
$$ \begin{aligned} & \left(13^3 \div 7^6\right)^2 \\ & =13^{3(2)} \div 7^{6(2)} \\ & =13^6 \div 7^{12} \end{aligned} $$
(d)
$$ \begin{aligned} & \left(5^3 \times 3^4\right)^5 \\ & =5^{3(5)} \times 3^{4(5)} \\ & =5^{15} \times 3^{20} \end{aligned} $$
(e)
$$ \begin{aligned} & \left(m^3 n^4 p^2\right)^5 \\ & =m^{3(5)} n^{4(5)} p^{2(5)} \\ & =m^{15} n^{20} p^{10} \end{aligned} $$
(f)
$$ \begin{aligned} & \left(2 w^2 x^3\right)^4 \\ & =2^{1(4)} w^{2(4)} x^{3(4)} \\ & =2^4 w^8 x^{12} \\ & =16 w^8 x^{12} \end{aligned} $$
(g)
$$ \begin{aligned} & \left(\frac{-3 a^5}{b^4}\right)^6 \\ & =\frac{-3^{1(6)} a^{5(6)}}{b^{4(6)}} \\ & =\frac{-729 a^{30}}{b^{24}} \end{aligned} $$
(h)
$$ \begin{aligned} & \left(\frac{2 a^5}{3 b^4}\right)^3 \\ & =\frac{2^{1(3)} a^{5(3)}}{3^{1(3)} b^{4(3)}} \\ & =\frac{8 a^{15}}{27 b^{12}} \end{aligned} $$
Ringkaskan setiap yang berikut.
Penyelesaian:
(a)
$$ \begin{aligned} & \left(2 \times 3^4\right)^2 \\ & =2^{1(2)} \times 3^{4(2)} \\ & =2^2 \times 3^8 \end{aligned} $$
(b)
$$ \begin{aligned} & \left(11^3 \times 9^5\right)^3 \\ & =11^{3(3)} \times 9^{5(3)} \\ & =11^9 \times 9^{15} \end{aligned} $$
(c)
$$ \begin{aligned} & \left(13^3 \div 7^6\right)^2 \\ & =13^{3(2)} \div 7^{6(2)} \\ & =13^6 \div 7^{12} \end{aligned} $$
(d)
$$ \begin{aligned} & \left(5^3 \times 3^4\right)^5 \\ & =5^{3(5)} \times 3^{4(5)} \\ & =5^{15} \times 3^{20} \end{aligned} $$
(e)
$$ \begin{aligned} & \left(m^3 n^4 p^2\right)^5 \\ & =m^{3(5)} n^{4(5)} p^{2(5)} \\ & =m^{15} n^{20} p^{10} \end{aligned} $$
(f)
$$ \begin{aligned} & \left(2 w^2 x^3\right)^4 \\ & =2^{1(4)} w^{2(4)} x^{3(4)} \\ & =2^4 w^8 x^{12} \\ & =16 w^8 x^{12} \end{aligned} $$
(g)
$$ \begin{aligned} & \left(\frac{-3 a^5}{b^4}\right)^6 \\ & =\frac{-3^{1(6)} a^{5(6)}}{b^{4(6)}} \\ & =\frac{-729 a^{30}}{b^{24}} \end{aligned} $$
(h)
$$ \begin{aligned} & \left(\frac{2 a^5}{3 b^4}\right)^3 \\ & =\frac{2^{1(3)} a^{5(3)}}{3^{1(3)} b^{4(3)}} \\ & =\frac{8 a^{15}}{27 b^{12}} \end{aligned} $$
Soalan 2:
Permudahkan setiap yang berikut.
Penyelesaian:
(a)
$$ \begin{aligned} & \left(\frac{11^3 \times 4^2}{11^2}\right)^2 \\ & =\frac{11^{3(2)} \times 4^{2(2)}}{11^{2(2)}} \\ & =\frac{11^6 \times 4^4}{11^4} \\ & =11^{6-4} \times 4^4 \\ & =11^2 \times 4^4 \end{aligned} $$
(b)
$$ \begin{aligned} & \frac{3^3 \times\left(6^2\right)^3}{6^4} \\ & =\frac{3^3 \times 6^6}{6^4} \\ & =3^3 \times 6^{6-4} \\ & =3^3 \times 6^2 \end{aligned} $$
(c)
$$ \begin{aligned} & \left(\frac{4^2}{6^3}\right)^3 \div \frac{4^2}{6^3} \\ & =\frac{4^6}{6^9} \div \frac{4^2}{6^3} \\ & =\frac{4^{6-2}}{6^{9-3}} \\ & =\frac{4^4}{6^6} \end{aligned} $$
(d)
$$ \begin{aligned} & \frac{\left[(-4)^6\right]^2 \times\left(-5^2\right)^3}{(-4)^6 \times(-5)^2} \\ & =\frac{(-4)^{12} \times(-5)^6}{(-4)^6 \times(-5)^2} \\ & =(-4)^{12-6} \times(-5)^{6-2} \\ & =(-4)^6 \times(-5)^4 \end{aligned} $$
(e)
$$ \begin{aligned} & \frac{x^2 y^6 \times x^3}{x y^2} \\ & =x^{2+3-1} y^{6-2} \\ & =x^4 y^4 \end{aligned} $$
(f)
$$ \begin{aligned} & \frac{\left(h^3 k^2\right)^4}{(h k)^2} \\ & =\frac{h^{12} k^8}{h^2 k^2} \\ & =h^{12-2} k^{8-2} \\ & =h^{10} k^6 \end{aligned} $$
(g)
$$ \begin{aligned} & \frac{\left(m^5 n^7\right)^3}{\left(m^2 n^3\right)^2} \\ & =\frac{m^{15} n^{21}}{m^4 n^6} \\ & =m^{15-4} n^{21-6} \\ & =m^{11} n^{15} \end{aligned} $$
(h)
$$ \begin{aligned} & \frac{\left(b^2 d^4\right)^3}{\left(b^2 d^3\right)^2} \\ & =\frac{b^6 d^{12}}{b^4 d^6} \\ & =b^{6-4} d^{12-6} \\ & =b^2 d^6 \end{aligned} $$
Permudahkan setiap yang berikut.
Penyelesaian:
(a)
$$ \begin{aligned} & \left(\frac{11^3 \times 4^2}{11^2}\right)^2 \\ & =\frac{11^{3(2)} \times 4^{2(2)}}{11^{2(2)}} \\ & =\frac{11^6 \times 4^4}{11^4} \\ & =11^{6-4} \times 4^4 \\ & =11^2 \times 4^4 \end{aligned} $$
(b)
$$ \begin{aligned} & \frac{3^3 \times\left(6^2\right)^3}{6^4} \\ & =\frac{3^3 \times 6^6}{6^4} \\ & =3^3 \times 6^{6-4} \\ & =3^3 \times 6^2 \end{aligned} $$
(c)
$$ \begin{aligned} & \left(\frac{4^2}{6^3}\right)^3 \div \frac{4^2}{6^3} \\ & =\frac{4^6}{6^9} \div \frac{4^2}{6^3} \\ & =\frac{4^{6-2}}{6^{9-3}} \\ & =\frac{4^4}{6^6} \end{aligned} $$
(d)
$$ \begin{aligned} & \frac{\left[(-4)^6\right]^2 \times\left(-5^2\right)^3}{(-4)^6 \times(-5)^2} \\ & =\frac{(-4)^{12} \times(-5)^6}{(-4)^6 \times(-5)^2} \\ & =(-4)^{12-6} \times(-5)^{6-2} \\ & =(-4)^6 \times(-5)^4 \end{aligned} $$
(e)
$$ \begin{aligned} & \frac{x^2 y^6 \times x^3}{x y^2} \\ & =x^{2+3-1} y^{6-2} \\ & =x^4 y^4 \end{aligned} $$
(f)
$$ \begin{aligned} & \frac{\left(h^3 k^2\right)^4}{(h k)^2} \\ & =\frac{h^{12} k^8}{h^2 k^2} \\ & =h^{12-2} k^{8-2} \\ & =h^{10} k^6 \end{aligned} $$
(g)
$$ \begin{aligned} & \frac{\left(m^5 n^7\right)^3}{\left(m^2 n^3\right)^2} \\ & =\frac{m^{15} n^{21}}{m^4 n^6} \\ & =m^{15-4} n^{21-6} \\ & =m^{11} n^{15} \end{aligned} $$
(h)
$$ \begin{aligned} & \frac{\left(b^2 d^4\right)^3}{\left(b^2 d^3\right)^2} \\ & =\frac{b^6 d^{12}}{b^4 d^6} \\ & =b^{6-4} d^{12-6} \\ & =b^2 d^6 \end{aligned} $$
Soalan 3:
Permudahkan setiap yang berikut.
Penyelesaian:
(a)
$$ \begin{aligned} & \frac{\left(2 m^2 n^4\right)^3 \times\left(3 m n^4\right)^2}{12 m^7 n^{12}} \\ & =\frac{2^3 m^6 n^{12} \times 3^2 m^2 n^8}{12 m^7 n^{12}} \\ & =\left(\frac{8 \times 9}{12}\right) m^{6+2-7} n^{12+8-12} \\ & =6 m n^8 \end{aligned} $$
(b)
$$ \begin{aligned} & \frac{\left(5 x y^4\right)^2 \times 6 x^{10} y}{15 x^4 y^6} \\ = & \frac{5^2 x^2 y^8 \times 6 x^{10} y}{15 x^4 y^6} \\ = & \left(\frac{25 \times 6}{15}\right) x^{2+10-4} y^{8+1-6} \\ = & 10 x^8 y^3 \end{aligned} $$
(c)
$$ \begin{aligned} \frac{24 d^3 e^5 \times\left(3 d^3 e^4\right)^2}{\left(d^5 e^6\right) \times\left(6 d e^2\right)^3} & =\frac{24 d^3 e^5 \times 3^2 d^6 e^8}{d^5 e^6 \times 216 d^3 e^6} \\ & =\left(\frac{24 \times 9}{216}\right) d^{3+6-(5+3)} e^{5+8-(6+6)} \\ & =d^{9-8} e^{13-12} \\ & =d e \end{aligned} $$
Permudahkan setiap yang berikut.
Penyelesaian:
(a)
$$ \begin{aligned} & \frac{\left(2 m^2 n^4\right)^3 \times\left(3 m n^4\right)^2}{12 m^7 n^{12}} \\ & =\frac{2^3 m^6 n^{12} \times 3^2 m^2 n^8}{12 m^7 n^{12}} \\ & =\left(\frac{8 \times 9}{12}\right) m^{6+2-7} n^{12+8-12} \\ & =6 m n^8 \end{aligned} $$
(b)
$$ \begin{aligned} & \frac{\left(5 x y^4\right)^2 \times 6 x^{10} y}{15 x^4 y^6} \\ = & \frac{5^2 x^2 y^8 \times 6 x^{10} y}{15 x^4 y^6} \\ = & \left(\frac{25 \times 6}{15}\right) x^{2+10-4} y^{8+1-6} \\ = & 10 x^8 y^3 \end{aligned} $$
(c)
$$ \begin{aligned} \frac{24 d^3 e^5 \times\left(3 d^3 e^4\right)^2}{\left(d^5 e^6\right) \times\left(6 d e^2\right)^3} & =\frac{24 d^3 e^5 \times 3^2 d^6 e^8}{d^5 e^6 \times 216 d^3 e^6} \\ & =\left(\frac{24 \times 9}{216}\right) d^{3+6-(5+3)} e^{5+8-(6+6)} \\ & =d^{9-8} e^{13-12} \\ & =d e \end{aligned} $$