### Theory:

Repeat!

 Grade Solution The value of the step ${10}^{3}$ = $10·10·10$ $$= 1000$$ ${10}^{2}$ = $10·10$ $$= 100$$ ${10}^{1}$ = any number in the first degreeis equal to itself $$= 10$$ ${10}^{0}$ = any number in the zero degreeis equal to 1 $$= 1$$ ${10}^{-1}$ = $\frac{1}{{10}^{1}}=\frac{1}{10}$ $$= 0.1$$ ${10}^{-2}$ = $\frac{1}{{10}^{2}}=\frac{1}{10\cdot 10}=\frac{1}{100}$ $$= 0.01$$ ${10}^{-3}$ = $\frac{1}{{10}^{3}}=\frac{1}{10\cdot 10\cdot 10}=\frac{1}{1000}$ $$= 0.001$$

 $\phantom{\rule{0.147em}{0ex}}{a}^{-n}=\frac{1}{{a}^{n}}$

To record very large or very small numbers, use the normal form.
The normal form of a number is called the multiple of this number: $a\cdot {10}^{n}\phantom{\rule{0.147em}{0ex}}\mathit{where}\phantom{\rule{0.147em}{0ex}}1\le a<10\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}$
Note that greater than $$1$$ and less than $$10.$$

If the number is greater than or equal to $$10,$$ then writes in normal form $$10$$ with a positive lever,
for example, blue whale mass is approx $1.9·{10}^{5}$$$kg$$.

If the number is less than $$1,$$ then writes in normal form $$10$$ with a negative gain,
for example, the mass of the smallest ants is approx $$0.000001 kg =$$ $1\cdot {10}^{-6}$ $$kg$$.

Remember that an integer is after the last digit.
 $$1 = 1.0$$ $$300 = 300.0$$ $$50,000 = 50,000.0$$ $$20 = 20.0$$ $$4000 = 4000.0$$ $$600,000 = 600,$$$$000.0$$

Important!
When switching from a number to a normal notation (or vice versa), move the period to the right or left in the number and multiply it by $$10$$ the appropriate degree.
If the normal format is to write a number greater than $$10,$$ then move the period to the left.
Example:
 $$98765 =$$ $9.8765·{10}^{4}$ Period moved $$4$$ places to the left. $\begin{array}{l}9.8765\overline{).}0\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}←\end{array}$ $$12345600 =$$ $1.23456·{10}^{7}$ Period moved $$7$$ places to the left. $\begin{array}{l}1.2345600\overline{).}0\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}←\end{array}$
If the normal form is to write a number smaller than $$1,$$ then move the period to the right.
Example:
 $$0.012345 =$$ $1.2345\cdot {10}^{-2}$ Period moved $$2$$ places to the right. $\begin{array}{l}0\overline{).}01.2345\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\to \end{array}$ $$0.00234567 =$$  $2.34567\cdot {10}^{-3}$ Period moved $$3$$ places to the right. $\begin{array}{l}0\overline{).}002.34567\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\to \end{array}$ $$0.000789 =$$$7.89\cdot {10}^{-4}$ Period moved $$4$$ places to the right. $\begin{array}{l}0\overline{).}0007.89\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\to \end{array}$
Reference:
Mathematics for 7th grade / Ilze France, Gunta Lace, Ligita Pickaine, Anita Mikelsone. - Lielvarde: Lielvārde, 2007. - 248 pp. - References: 119-120. p.