Question

Write the following in scientific notation.
(i) ${\left(4000000\right)}^3$
(ii) ${\left(5000\right)}^4\times {\left(200\right)}^3$
(iii) ${\left(0.00003\right)}^5$
(iv) ${\left(2000\right)}^2÷{\left(0.0001\right)}^4$

Answer 1

(i) First we write the number (within the brackets) in scientific notation.
$4000000=4.0\times {10}^6$
Now, raising to the power $3$ on both sides we get,
$\therefore {\left(4000000\right)}^3={(4.0\times {10}^6)}^3={\left(4.0\right)}^3\times {\left({10}^6\right)}^3$
$=64\times {10}^{18}=6.4\times {10}^1\times {10}^{18}=6.4\times {10}^{19}$
(ii) In scientific notation,
$5000=5.0\times {10}^3$ and $200=2.0\times {10}^2$
$\therefore {\left(5000\right)}^4\times {\left(200\right)}^3={(5.0\times {10}^3)}^4{(2.0\times {10}^2)}^3$
$={\left(5.0\right)}^4\times {\left({10}^3\right)}^4\times {\left(2.0\right)}^3\times {\left({10}^2\right)}^3$
$=625\times {10}^{12}\times 8\times {10}^6=5000\times {10}^{18}$
$=5.0\times {10}^3\times {10}^{18}=5.0\times {10}^{21}$
(iii) In scientific notation, $0.00003=3.0\times {10}^{-5}$
$\therefore {\left(0.00003\right)}^5={(3.0\times {10}^{-5})}^5={\left(3.0\right)}^5\times {\left({10}^{-5}\right)}^5$
$=243\times {10}^{-25}=2.43\times {10}^2\times {10}^{-25}=2.43\times {10}^{-23}$
(iv) In scientific notation,
$2000=2.0\times {10}^3$ and $0.0001=1.0\times {10}^{-4}$
$\therefore {\left(2000\right)}^2÷{\left(0.0001\right)}^4=\frac{{(2.0\times {10}^3)}^2}{{(1.0\times {10}^{-4})}^4}=\frac{{\left(2.0\right)}^2\times {\left({10}^3\right)}^2}{{\left(1.0\right)}^4\times {\left({10}^{-4}\right)}^4}$
$=\frac{4\times {10}^6}{{10}^{-16}}=4.0\times {10}^{6-(-16)}=4.0\times {10}^{22}$