A number with an exponent is called a power. Here, you will learn about powers of 10. A power of 10 is a power that has $10$ as the base number (ground floor).
The exponent (attic) shows how many times $10$ should be multiplied by itself.
Rule
$$1{0}^{n}=\underset{\text{}n\text{times}}{\underset{\u23df}{10\cdot 10\cdots 10}}$$ |
$10$ is the base number, and $n$ is the exponent.
Example 1
What does $1{0}^{4}$ mean?
From the power, you can see that the base number is $10$ and that the exponent is $4$. Therefore, $10$ has to be multiplied with itself $4$ times:
$$1{0}^{4}=10\cdot 10\cdot 10\cdot 10=10\phantom{\rule{0.17em}{0ex}}000$$ |
What does $1{0}^{7}$ mean?
From the power, you can see that the base number is $10$ and that the exponent is $7$. Therefore, $10$ has to be multiplied with itself $7$ times:
$$1{0}^{7}=10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10=10\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}000$$ |
Rule
$$1{0}^{-n}=\frac{1}{1{0}^{n}}$$ |
The formula above says that you can change the floor of the power by changing the sign of the exponent. It does not matter whether you change the sign from minus to plus or from plus to minus, as long as it changes. Below, you will see some examples of how this formula works with powers of 10.
What does $1{0}^{-7}$ mean? It means that the power is put in the denominator of a fraction with $1$ as the numerator, while removing the minus sign in the exponent. Then, the base number $10$ is multiplied by itself $7$ times. It is the minus sign of the exponent that tells you to rewrite it into a fraction to make the exponent positive.
$$1{0}^{-7}=\frac{1}{1{0}^{7}}=\frac{1}{10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10\cdot 10}=0.000\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{0.17em}{0ex}}1$$ |
Example 2
$$\begin{array}{llll}\hfill 1{0}^{-1}& =\frac{1}{1{0}^{1}}=\frac{1}{10}=0.1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 1{0}^{-3}& =\frac{1}{1{0}^{3}}=\frac{1}{10\cdot 10\cdot 10}=0.001\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{1}{1{0}^{-5}}& =1{0}^{5}=10\cdot 10\cdot 10\cdot 10\cdot 10\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =100\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
Rule
When you work with powers of 10, we have the following rules:
When multiplying by $10$, you move the point one place to the right.
When multiplying by $1{0}^{n}$, you move the point $n$ places to the right.
When multiplying by $1{0}^{-1}$, you move the point one place to the left.
When multiplying by $1{0}^{-n}$, you move the point $n$ places to the left.
Example 3
$$1.3\cdot 10=13\phantom{\rule{2em}{0ex}}26.49\cdot 1{0}^{3}=26\phantom{\rule{0.17em}{0ex}}490$$ |
$$1.3\cdot 10=13\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}26.49\cdot 1{0}^{3}=26\phantom{\rule{0.17em}{0ex}}490$$ |
Example 4
$$1.3\cdot 1{0}^{-1}=0.13\phantom{\rule{2em}{0ex}}26.49\cdot 1{0}^{-3}=0.026\phantom{\rule{0.17em}{0ex}}49$$ |
$$1.3\cdot 1{0}^{-1}=0.13\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}26.49\cdot 1{0}^{-3}=0.026\phantom{\rule{0.17em}{0ex}}49$$ |
If you want to practice multiplying by the power of 10, I recommend that you watch instructional videos about multiplying by the power of 10.