# Powers of 10

When we talked about exponents, we said that raising $a$ to $b$-th power is $a^b=\underbrace{a\cdot a\cdot a\cdot a\cdot...\cdot a}_{b}$.

$a$ is called base, $b$ is exponent (power).

We assumed that $b$ is positive integer.

Here we will talk about special case $10^b$, i.e. when the base equals $10$.

So, for example, ${{10}}^{{3}}={10}\cdot{10}\cdot{10}={1000}$ and ${{10}}^{{4}}={10}\cdot{10}\cdot{10}\cdot{10}={10000}$.

We can see a pattern here. Exponent shows how many zeros will be there in result.

${{10}}^{{{\color{red}{{{7}}}}}}={1}{\color{red}{{{0000000}}}}$ : exponent equals 7, so in result there will be 1 with seven zeros.

This makes it easier to calculate powers of ten.

Now, let's see how powers of 10 are related to decimals. For this we need to recall negative exponents.

So, if ${b}$ is positive integer then ${{a}}^{{-{b}}}=\frac{{1}}{{{{a}}^{{b}}}}$.

In case of 10 we have that ${{10}}^{{-{b}}}=\frac{{1}}{{{{10}}^{{b}}}}$.

For example, ${{10}}^{{-{2}}}=\frac{{1}}{{{{10}}^{{2}}}}=\frac{{1}}{{100}}$.

There is special notation for such fraction: $\frac{{1}}{{100}}={0.01}$.

In general, ${{10}}^{{-{b}}}$ is a decimal that has ${b}$ digits in decimal part and all digits except last equals 0. Last digit is 1.

For example, ${{10}}^{{-{\color{red}{{{4}}}}}}={0}.{\color{red}{{{0001}}}}$ (4 digits in decimal part; all zeros except last).

These things relate decimals and mixed numbers, meaning that mixed number is a decimal written in another form and vice versa.

Rule. If we multiply by 10, we move decimal point one postion to the right, if we divide by 10, we move decimal point to the left.

Suppose, you have number 500. It can be written as ${50}\cdot{10}={50}\times{{10}}^{{1}}$ or it can be written as ${5}\cdot{100}={5}\times{{10}}^{{2}}$.

We can even further move decimal point to the left and write, that ${500}={0.5}\cdot{1000}={0.5}\times{{10}}^{{3}}$.

Same works with negative exponent ${0.7}={7}\cdot\frac{{1}}{{10}}={7}\times{{10}}^{{-{1}}}={70}\cdot\frac{{1}}{{100}}={7}\times{{10}}^{{-{2}}}$.

Moving in another direction is also applicable: ${0.7}={0.07}\cdot{10}={0.07}\times{{10}}^{{1}}$.

Now, let's practice a bit.

Exercise 1. Convert ${{10}}^{{-{{3}}}}$ to decimal.

Answer: ${{10}}^{{-{{3}}}}={0.001}$.

Next exercise.

Exercise 2. Convert ${{10}}^{{-{5}}}$ to decimal.

Answer: ${{10}}^{{-{5}}}={0.00001}$.

Last exercise.

Exercise 3. Convert ${0.0001}$ into fraction.

Answer: ${0.0001}=\frac{{1}}{{{10000}}}={{10}}^{{-{4}}}$.