Soalan 2:
Diberi m = 3.2 × 103 dan n = 5.43 × 104. Hitung nilai operasi berikut. Nyatakan jawapan anda dalam bentuk piawai dan betul kepada tiga angka bererti.
(a) 2 mn
(b) m + n
(c) n – m
(d) m2 + n2
(e) 3m/2n
(f) m + 2/mn
(g) m–2 + n–3
(h) n – m–3
Penyelesaian:
(a)
$$ \begin{aligned} & 2 m n \\ & =2 \times 3.2 \times 10^3 \times 5.43 \times 10^4 \\ & =2 \times 3.2 \times 5.43 \times 10^{3+4} \\ & =34.752 \times 10^7 \\ & =3.4752 \times 10^{7+1} \\ & =3.48 \times 10^8(3 \text { a.b. }) \end{aligned} $$
(b)
$$ \begin{aligned} & m+n \\ & =3.2 \times 10^3+5.43 \times 10^4 \\ & =0.32 \times 10^{3+1}+5.43 \times 10^4 \\ & =(0.32+5.43) \times 10^4 \\ & =5.75 \times 10^4[3 \mathrm{a} . \mathrm{b} .] \end{aligned} $$
(c)
$$ \begin{aligned} & n-m \\ & =5.43 \times 10^4-3.2 \times 10^3 \\ & =5.43 \times 10^4-0.32 \times 10^{3+1} \\ & =(5.43-0.32) \times 10^4 \\ & =5.11 \times 10^4(3 \mathrm{a} . \mathrm{b} .) \end{aligned} $$
(d)
$$ \begin{aligned} & m^2+n^2 \\ & =\left(3.2 \times 10^3\right)^2+\left(5.43 \times 10^4\right)^2 \\ & =3.2^2 \times 10^6+5.43^2 \times 10^8 \\ & =10.24 \times 10^6+29.4849 \times 10^8 \\ & =0.1024 \times 10^{6+2}+29.4849 \times 10^8 \\ & =(0.1024+29.4849) \times 10^8 \\ & =29.5873 \times 10^8 \\ & =2.95873 \times 10^{8+1} \\ & =2.96 \times 10^9[3 \text { a.b. }] \end{aligned} $$
(e)
$$ \begin{aligned} \frac{3 m}{2 n} & =\frac{3\left(3.2 \times 10^3\right)}{2\left(5.43 \times 10^4\right)} \\ & =\frac{9.6 \times 10^3}{10.86 \times 10^4} \\ & =\frac{9.6}{10.86} \times 10^{3-4} \\ & =0.884 \times 10^{-1} \\ & =8.84 \times 10^{-1-1} \\ & =8.84 \times 10^{-2}[3 \text { a.b. }] \end{aligned} $$
(f)
$$ \begin{aligned} \frac{m+n}{m n} & =\frac{3.2 \times 10^3+5.43 \times 10^4}{3.2 \times 10^3 \times 5.43 \times 10^4} \\ & =\frac{3.2 \times 10^3+54.3 \times 10^{4-1}}{3.2 \times 5.43 \times 10^{3+4}} \\ & =\frac{(3.2+54.3) \times 10^3}{17.376 \times 10^7} \\ & =\frac{57.5 \times 10^3}{17.376 \times 10^7} \\ & =\frac{57.5}{17.376} \times 10^{3-7} \\ & =3.3092 \times 10^{-4} \\ & =3.31 \times 10^{-4}[3 \mathrm{a} . \mathrm{b} .] \end{aligned} $$
(g)
$$ \begin{aligned} m^{-2}+n^{-3} & =\left(3.2 \times 10^3\right)^{-2}+\left(5.43 \times 10^4\right)^{-3} \\ & =3.2^{-2} \times 10^{(3)(-2)}+5.43^{-3} \times 10^{(4 \times-3)} \\ & =\frac{1}{3.2^2} \times 10^{-6}+\frac{1}{5.43^3} \times 10^{-12} \\ & =0.097656 \times 10^{-6}+0.006246 \times 10^{-12} \\ & =0.097656 \times 10^{-6}+0.000000006246 \times 10^{-6} \\ & =0.0977 \times 10^{-6} \\ & =9.77 \times 10^{-6-2} \\ & =9.77 \times 10^{-8}[3 \text { a.b. }] \end{aligned} $$
(h)
$$ \begin{aligned} &\begin{aligned} & n-m^{-3} \\ & =5.43 \times 10^4-\left(3.2 \times 10^3\right)^{-3} \\ & =5.43 \times 10^4-3.2^{-3} \times 10^{(3 \times-3)} \\ & =5.43 \times 10^4-\frac{1}{3.2^3} \times 10^{-9} \\ & =5.43 \times 10^4-0.03052 \times 10^{-9} \end{aligned}\\ &\text { (angka yang terlalu kecil akan diabaikan) }\\ &=5.43 \times 10^4[3 \text { a.b. }] \end{aligned} $$
Diberi m = 3.2 × 103 dan n = 5.43 × 104. Hitung nilai operasi berikut. Nyatakan jawapan anda dalam bentuk piawai dan betul kepada tiga angka bererti.
(a) 2 mn
(b) m + n
(c) n – m
(d) m2 + n2
(e) 3m/2n
(f) m + 2/mn
(g) m–2 + n–3
(h) n – m–3
Penyelesaian:
(a)
$$ \begin{aligned} & 2 m n \\ & =2 \times 3.2 \times 10^3 \times 5.43 \times 10^4 \\ & =2 \times 3.2 \times 5.43 \times 10^{3+4} \\ & =34.752 \times 10^7 \\ & =3.4752 \times 10^{7+1} \\ & =3.48 \times 10^8(3 \text { a.b. }) \end{aligned} $$
(b)
$$ \begin{aligned} & m+n \\ & =3.2 \times 10^3+5.43 \times 10^4 \\ & =0.32 \times 10^{3+1}+5.43 \times 10^4 \\ & =(0.32+5.43) \times 10^4 \\ & =5.75 \times 10^4[3 \mathrm{a} . \mathrm{b} .] \end{aligned} $$
(c)
$$ \begin{aligned} & n-m \\ & =5.43 \times 10^4-3.2 \times 10^3 \\ & =5.43 \times 10^4-0.32 \times 10^{3+1} \\ & =(5.43-0.32) \times 10^4 \\ & =5.11 \times 10^4(3 \mathrm{a} . \mathrm{b} .) \end{aligned} $$
(d)
$$ \begin{aligned} & m^2+n^2 \\ & =\left(3.2 \times 10^3\right)^2+\left(5.43 \times 10^4\right)^2 \\ & =3.2^2 \times 10^6+5.43^2 \times 10^8 \\ & =10.24 \times 10^6+29.4849 \times 10^8 \\ & =0.1024 \times 10^{6+2}+29.4849 \times 10^8 \\ & =(0.1024+29.4849) \times 10^8 \\ & =29.5873 \times 10^8 \\ & =2.95873 \times 10^{8+1} \\ & =2.96 \times 10^9[3 \text { a.b. }] \end{aligned} $$
(e)
$$ \begin{aligned} \frac{3 m}{2 n} & =\frac{3\left(3.2 \times 10^3\right)}{2\left(5.43 \times 10^4\right)} \\ & =\frac{9.6 \times 10^3}{10.86 \times 10^4} \\ & =\frac{9.6}{10.86} \times 10^{3-4} \\ & =0.884 \times 10^{-1} \\ & =8.84 \times 10^{-1-1} \\ & =8.84 \times 10^{-2}[3 \text { a.b. }] \end{aligned} $$
(f)
$$ \begin{aligned} \frac{m+n}{m n} & =\frac{3.2 \times 10^3+5.43 \times 10^4}{3.2 \times 10^3 \times 5.43 \times 10^4} \\ & =\frac{3.2 \times 10^3+54.3 \times 10^{4-1}}{3.2 \times 5.43 \times 10^{3+4}} \\ & =\frac{(3.2+54.3) \times 10^3}{17.376 \times 10^7} \\ & =\frac{57.5 \times 10^3}{17.376 \times 10^7} \\ & =\frac{57.5}{17.376} \times 10^{3-7} \\ & =3.3092 \times 10^{-4} \\ & =3.31 \times 10^{-4}[3 \mathrm{a} . \mathrm{b} .] \end{aligned} $$
(g)
$$ \begin{aligned} m^{-2}+n^{-3} & =\left(3.2 \times 10^3\right)^{-2}+\left(5.43 \times 10^4\right)^{-3} \\ & =3.2^{-2} \times 10^{(3)(-2)}+5.43^{-3} \times 10^{(4 \times-3)} \\ & =\frac{1}{3.2^2} \times 10^{-6}+\frac{1}{5.43^3} \times 10^{-12} \\ & =0.097656 \times 10^{-6}+0.006246 \times 10^{-12} \\ & =0.097656 \times 10^{-6}+0.000000006246 \times 10^{-6} \\ & =0.0977 \times 10^{-6} \\ & =9.77 \times 10^{-6-2} \\ & =9.77 \times 10^{-8}[3 \text { a.b. }] \end{aligned} $$
(h)
$$ \begin{aligned} &\begin{aligned} & n-m^{-3} \\ & =5.43 \times 10^4-\left(3.2 \times 10^3\right)^{-3} \\ & =5.43 \times 10^4-3.2^{-3} \times 10^{(3 \times-3)} \\ & =5.43 \times 10^4-\frac{1}{3.2^3} \times 10^{-9} \\ & =5.43 \times 10^4-0.03052 \times 10^{-9} \end{aligned}\\ &\text { (angka yang terlalu kecil akan diabaikan) }\\ &=5.43 \times 10^4[3 \text { a.b. }] \end{aligned} $$