Soalan 1:
Nyatakan setiap sebutan berikut dalam bentuk indeks positif.
Penyelesaian:
(a)
$$ 5^{-3}=\frac{1}{5^3} $$
(b)
$$ 8^{-4}=\frac{1}{8^4} $$
(c)
$$ x^{-8}=\frac{1}{x^8} $$
(d)
$$ y^{-16}=\frac{1}{y^{16}} $$
(e)
$$ \frac{1}{a^{-4}}=a^4 $$
(f)
$$ \frac{1}{20^{-2}}=20^2 $$
(g)
$$ 3 n^{-4}=3\left(\frac{1}{n^4}\right)=\frac{3}{n^4} $$
(h)
$$ -5 n^{-6}=-5\left(\frac{1}{n^6}\right)=-\frac{5}{n^6} $$
(i)
$$ \frac{2}{7} m^{-5}=\frac{2}{7}\left(\frac{1}{m^5}\right)=\frac{2}{7 m^5} $$
(j)
$$ \left(-\frac{3}{8}\right) m^{-4}=\left(-\frac{3}{8}\right)\left(\frac{1}{m^4}\right)=-\frac{3}{8 m^4} $$
(k)
$$ \left(\frac{2}{5}\right)^{-12}=\left(\frac{5}{2}\right)^{12} $$
(l)
$$ \left(-\frac{3}{7}\right)^{-14}=\left(-\frac{7}{3}\right)^{14} $$
(m)
$$ \left(\frac{x}{y}\right)^{-10}=\left(\frac{y}{x}\right)^{10} $$
(n)
$$ \left(\frac{2 x}{3 y}\right)^{-4}=\left(\frac{3 y}{2 x}\right)^4 $$
(o)
$$ \left(\frac{1}{2 x}\right)^{-5}=(2 x)^5 $$
Nyatakan setiap sebutan berikut dalam bentuk indeks positif.
Penyelesaian:
(a)
$$ 5^{-3}=\frac{1}{5^3} $$
(b)
$$ 8^{-4}=\frac{1}{8^4} $$
(c)
$$ x^{-8}=\frac{1}{x^8} $$
(d)
$$ y^{-16}=\frac{1}{y^{16}} $$
(e)
$$ \frac{1}{a^{-4}}=a^4 $$
(f)
$$ \frac{1}{20^{-2}}=20^2 $$
(g)
$$ 3 n^{-4}=3\left(\frac{1}{n^4}\right)=\frac{3}{n^4} $$
(h)
$$ -5 n^{-6}=-5\left(\frac{1}{n^6}\right)=-\frac{5}{n^6} $$
(i)
$$ \frac{2}{7} m^{-5}=\frac{2}{7}\left(\frac{1}{m^5}\right)=\frac{2}{7 m^5} $$
(j)
$$ \left(-\frac{3}{8}\right) m^{-4}=\left(-\frac{3}{8}\right)\left(\frac{1}{m^4}\right)=-\frac{3}{8 m^4} $$
(k)
$$ \left(\frac{2}{5}\right)^{-12}=\left(\frac{5}{2}\right)^{12} $$
(l)
$$ \left(-\frac{3}{7}\right)^{-14}=\left(-\frac{7}{3}\right)^{14} $$
(m)
$$ \left(\frac{x}{y}\right)^{-10}=\left(\frac{y}{x}\right)^{10} $$
(n)
$$ \left(\frac{2 x}{3 y}\right)^{-4}=\left(\frac{3 y}{2 x}\right)^4 $$
(o)
$$ \left(\frac{1}{2 x}\right)^{-5}=(2 x)^5 $$
Soalan 2:
Nyatakan setiap sebutan berikut dalam bentuk indeks negatif.
Penyelesaian:
(a)
$$ \frac{1}{5^4}=5^{-4} $$
(b)
$$ \frac{1}{8^3}=8^{-3} $$
(c)
$$ \frac{1}{m^7}=m^{-7} $$
(d)
$$ \frac{1}{n^9}=n^{-9} $$
(e)
$$ 10^2=\frac{1}{10^{-2}} $$
(f)
$$ (-4)^3=\frac{1}{(-4)^{-3}} $$
(g)
$$ m^{12}=\frac{1}{m^{-12}} $$
(h)
$$ n^{16}=\frac{1}{n^{-16}} $$
(i)
$$ \left(\frac{4}{7}\right)^9=\left(\frac{7}{4}\right)^{-9} $$
(j)
$$ \left(\frac{x}{y}\right)^{10}=\left(\frac{y}{x}\right)^{-10} $$
Nyatakan setiap sebutan berikut dalam bentuk indeks negatif.
Penyelesaian:
(a)
$$ \frac{1}{5^4}=5^{-4} $$
(b)
$$ \frac{1}{8^3}=8^{-3} $$
(c)
$$ \frac{1}{m^7}=m^{-7} $$
(d)
$$ \frac{1}{n^9}=n^{-9} $$
(e)
$$ 10^2=\frac{1}{10^{-2}} $$
(f)
$$ (-4)^3=\frac{1}{(-4)^{-3}} $$
(g)
$$ m^{12}=\frac{1}{m^{-12}} $$
(h)
$$ n^{16}=\frac{1}{n^{-16}} $$
(i)
$$ \left(\frac{4}{7}\right)^9=\left(\frac{7}{4}\right)^{-9} $$
(j)
$$ \left(\frac{x}{y}\right)^{10}=\left(\frac{y}{x}\right)^{-10} $$
Soalan 3:
Permudahkan setiap yang berikut.
Penyelesaian:
(a)
$$ \begin{aligned} & \frac{\left(4^2\right)^3 \times 4^5}{\left(4^6\right)^2} \\ & =\frac{4^{2 \times 3} \times 4^5}{4^{6 \times 2}} \\ & =\frac{4^6 \times 4^5}{4^{12}} \\ & =4^{6+5-12} \\ & =4^{-1} \\ & =\frac{1}{4} \end{aligned} $$
(b)
$$ \begin{aligned} & \frac{\left(2^3 \times 3^2\right)^3}{\left(2 \times 3^4\right)^5} \\ & =\frac{2^{3 \times 3} \times 3^{2 \times 3}}{2^{1 \times 5} \times 3^{4 \times 5}} \\ & =\frac{2^9 \times 3^6}{2^5 \times 3^{20}} \\ & =2^{9-5} \times 3^{6-20} \\ & =2^4 \times 3^{-14} \\ & =2^4 \times \frac{1}{3^{14}} \\ & =\frac{2^4}{3^{14}} \end{aligned} $$
(c)
$$ \begin{aligned} & \frac{\left(5^2\right)^5}{\left(2^3\right)^{-2} \times\left(5^4\right)^2} \\ & =\frac{5^{2 \times 5}}{2^{3 \times-2} \times 5^{4 \times 2}} \\ & =\frac{5^{10}}{2^{-6} \times 5^8} \\ & =\frac{5^{10-8}}{2^{-6}} \\ & =5^2 \times \frac{1}{2^{-6}} \\ & =5^2 \times 2^6 \\ & =2^6 \times 5^2 \end{aligned} $$
(d)
$$ \begin{aligned} & \frac{3 m^2 n^4 \times\left(m n^3\right)^{-2}}{9 m^3 n^5} \\ & =\frac{3 m^2 n^4 \times m^{1 \times-2} n^{3 \times-2}}{9 m^3 n^5} \\ & =\frac{3 m^2 n^4 \times m^{-2} n^{-6}}{9 m^3 n^5} \\ & =\frac{3}{9} \times m^{2+(-2)-3} \times n^{4+(-6)-5} \\ & =\frac{1}{3} \times m^{-3} n^{-7} \\ & =\frac{1}{3} \times \frac{1}{m^3} \times \frac{1}{n^7} \\ & =\frac{1}{3 m^3 n^7} \end{aligned} $$
(e)
$$ \begin{aligned} \frac{\left(2 m^2 n^2\right)^{-3} \times\left(3 m n^2\right)^4}{\left(9 m^3 n\right)^2} & =\frac{2^{-3} m^{-6} n^{-6} \times 3^4 m^4 n^8}{9^2 m^6 n^2} \\ & =\frac{2^{-3} \times 3^4}{9^2} \times m^{-6+4-6} \times n^{-6+8-2} \\ & =\frac{3^4}{9^2 \times 2^3} \times m^{-8} \times n^0 \\ & =\frac{3^4}{\left(3^2\right)^2 \times 2^3} \times \frac{1}{m^8} \quad n^0=1 \\ & =\frac{3^4}{3^4 \times 2^3} \times \frac{1}{m^8} \\ & =3^{4-4} \times \frac{1}{2^3} \times \frac{1}{m^8} \\ & =3^0 \times \frac{1}{2^3} \times \frac{1}{m^8} \\ & =\frac{1}{8 m^8} \end{aligned} $$
(f)
$$ \begin{aligned} \frac{\left(4 m^2 n^4\right)^2}{\left(2 m^{-2} n\right)^5 \times\left(3 m^4 n\right)^2} & =\frac{4^2 m^4 n^8}{2^5 m^{-10} n^5 \times 3^2 m^8 n^2} \\ & =\frac{4^2}{2^5 \times 3^2} \times m^{4-(-10)-8} \times n^{8-5-2} \\ & =\frac{2^{4-5}}{3^2} \times m^6 \times n^1 \\ & =\frac{2^{-1}}{3^2} \times m^6 \times n \\ & =\frac{1}{2 \times 9} m^6 n \\ & =\frac{1}{18} m^6 n \\ & =\frac{m^6 n}{18} \end{aligned} $$
Permudahkan setiap yang berikut.
Penyelesaian:
(a)
$$ \begin{aligned} & \frac{\left(4^2\right)^3 \times 4^5}{\left(4^6\right)^2} \\ & =\frac{4^{2 \times 3} \times 4^5}{4^{6 \times 2}} \\ & =\frac{4^6 \times 4^5}{4^{12}} \\ & =4^{6+5-12} \\ & =4^{-1} \\ & =\frac{1}{4} \end{aligned} $$
(b)
$$ \begin{aligned} & \frac{\left(2^3 \times 3^2\right)^3}{\left(2 \times 3^4\right)^5} \\ & =\frac{2^{3 \times 3} \times 3^{2 \times 3}}{2^{1 \times 5} \times 3^{4 \times 5}} \\ & =\frac{2^9 \times 3^6}{2^5 \times 3^{20}} \\ & =2^{9-5} \times 3^{6-20} \\ & =2^4 \times 3^{-14} \\ & =2^4 \times \frac{1}{3^{14}} \\ & =\frac{2^4}{3^{14}} \end{aligned} $$
(c)
$$ \begin{aligned} & \frac{\left(5^2\right)^5}{\left(2^3\right)^{-2} \times\left(5^4\right)^2} \\ & =\frac{5^{2 \times 5}}{2^{3 \times-2} \times 5^{4 \times 2}} \\ & =\frac{5^{10}}{2^{-6} \times 5^8} \\ & =\frac{5^{10-8}}{2^{-6}} \\ & =5^2 \times \frac{1}{2^{-6}} \\ & =5^2 \times 2^6 \\ & =2^6 \times 5^2 \end{aligned} $$
(d)
$$ \begin{aligned} & \frac{3 m^2 n^4 \times\left(m n^3\right)^{-2}}{9 m^3 n^5} \\ & =\frac{3 m^2 n^4 \times m^{1 \times-2} n^{3 \times-2}}{9 m^3 n^5} \\ & =\frac{3 m^2 n^4 \times m^{-2} n^{-6}}{9 m^3 n^5} \\ & =\frac{3}{9} \times m^{2+(-2)-3} \times n^{4+(-6)-5} \\ & =\frac{1}{3} \times m^{-3} n^{-7} \\ & =\frac{1}{3} \times \frac{1}{m^3} \times \frac{1}{n^7} \\ & =\frac{1}{3 m^3 n^7} \end{aligned} $$
(e)
$$ \begin{aligned} \frac{\left(2 m^2 n^2\right)^{-3} \times\left(3 m n^2\right)^4}{\left(9 m^3 n\right)^2} & =\frac{2^{-3} m^{-6} n^{-6} \times 3^4 m^4 n^8}{9^2 m^6 n^2} \\ & =\frac{2^{-3} \times 3^4}{9^2} \times m^{-6+4-6} \times n^{-6+8-2} \\ & =\frac{3^4}{9^2 \times 2^3} \times m^{-8} \times n^0 \\ & =\frac{3^4}{\left(3^2\right)^2 \times 2^3} \times \frac{1}{m^8} \quad n^0=1 \\ & =\frac{3^4}{3^4 \times 2^3} \times \frac{1}{m^8} \\ & =3^{4-4} \times \frac{1}{2^3} \times \frac{1}{m^8} \\ & =3^0 \times \frac{1}{2^3} \times \frac{1}{m^8} \\ & =\frac{1}{8 m^8} \end{aligned} $$
(f)
$$ \begin{aligned} \frac{\left(4 m^2 n^4\right)^2}{\left(2 m^{-2} n\right)^5 \times\left(3 m^4 n\right)^2} & =\frac{4^2 m^4 n^8}{2^5 m^{-10} n^5 \times 3^2 m^8 n^2} \\ & =\frac{4^2}{2^5 \times 3^2} \times m^{4-(-10)-8} \times n^{8-5-2} \\ & =\frac{2^{4-5}}{3^2} \times m^6 \times n^1 \\ & =\frac{2^{-1}}{3^2} \times m^6 \times n \\ & =\frac{1}{2 \times 9} m^6 n \\ & =\frac{1}{18} m^6 n \\ & =\frac{m^6 n}{18} \end{aligned} $$