Unit 7 Exponents And Scientific Notation — Unit Plan

In this lesson, we developed a rule for multiplying powers of 10: Multiplying powers of 10 corresponds to adding the exponents together. 

To see this, multiply $10^2$ and $10^3$. We know that $10^2$ has two factors that are 10 and that $10^3$ has three factors that are 10. That means that $10^2\boldcdot10^3$ has 5 factors that are 10. 

This will work for other powers of 10, too. For example, $10^{14} \boldcdot 10^{47} = 10^{(14+47)}=10^{61}$.

In this lesson, we developed a rule for dividing powers of 10: Dividing powers of 10 is the same as subtracting the exponent of the denominator from the exponent of the numerator. To see this, take $10^5$ and divide it by $10^2$. 

We know that $10^5$ has 5 factors that are 10, and 2 of these factors can be divided by the 2 factors of 10 in $10^2$ to make 1. That leaves $5-2=3$ factors of 10, or $10^3$.

This will work for other powers of 10, too. For example $\frac{10^{56}}{10^{23}}=10^{56-23}=10^{33}$.

This rule also extends to $10^0$. If we look at $\frac{10^6}{10^0}$, using the exponent rule gives $10^{6-0}$, which is equal to $10^6$. So dividing $10^6$ by $10^0$doesn’t change its value. That means if we want the rule to work when the exponent is 0, then $10^0$ must equal 1.

In this lesson, we observed that when we multiply a positive power of 10 by $\frac{1}{10}$, the exponent decreases by 1. For example, $10^8 \boldcdot \frac{1}{10} = 10^7$. This is true for any power of 10. 

By using the rule $10^n\boldcdot 10^m=10^{n+m}$ with this example, we see that: $10^8\boldcdot 10^{\text-1}=10^7$.

Notice that for the exponent rules we have developed to work, then $\frac{1}{10}$ must equal $10^{\text-1}$.

We can keep track of repeated factors using exponent rules. These rules also help us make sense of negative exponents and why a number to the power of 0 is defined as 1. These rules can be written symbolically where the base $a$ can be any positive number:

Sometimes powers of 10 are helpful for expressing quantities, especially very large or very small quantities.

For example, the United States Mint has made over 500,000,000,000 pennies. To understand this number we can look at the number of zeros to know it is equivalent to 500 billion pennies. Since 1 billion can be written as $10^9$, we can say that there are over $500\boldcdot 10^9$ pennies. 

Sometimes we may need to rewrite a number using a different power of 10. We can say that $500\boldcdot 10^9 = 5\boldcdot 10^{11}$. Since the factor $10^9$ was multiplied by 100 to get $10^{11}$, the factor of 500 was divided by 100 to keep the value of the entire expression the same.

The same is true for very small quantities. For example, a single atom of carbon weighs about 0.0000000000000000000000199 grams. If we write this as a fraction we get $\dfrac{199}{10,000,000,000,000,000,000,000,000}$. Using powers of 10, it becomes $199\boldcdot10^{\text-25}$, which is a lot easier to write!

Just as we did with large numbers, small numbers can be rewritten as an equivalent value with a different power of 10. In this example we can divide the factor 199 by 100 and multiply the factor $10^{\text-25}$ by 100 to get $1.99\boldcdot10^{\text-23}$.

The total value of all the quarters made in 2014 was 400 million dollars. There are many ways to express this using powers of 10. We could write this as $400 \boldcdot 10^6$ dollars, $40 \boldcdot 10^7$ dollars, $0.4 \boldcdot 10^9$ dollars, or many other ways. One special way to write this quantity is called scientific notation, where the first factor is a number greater than or equal to 1, but less than 10, and the second factor is an integer power of 10

In scientific notation,

400 million dollars

would be written as

$4 \times 10^8$ dollars.

Writing the number this way shows exactly where it lies between two consecutive powers of 10. The $10^8$ shows us the number is between $10^8$ and $10^9$. The 4 shows us that the number is 4 tenths of the way to $10^9$.

A number line, 11 tick marks, 0, 1 times 10 to the power 11, 2  times 10 to the power 11, 3 times 10 to the power 11, 4 times 10 to the power 11, 5 times 10 to the power 11, 6 times 10 to the power 11, 7 times 10 to the power 11, 8 times 10 to the power 11, 9 times 10 to the power 11, 10 to the power 12. Three times 10 to the power 11 to 4 times 10 to the power 11 is zoomed out, to 11 tick marks labeled 3 times 10 to the power 11, blank, blank, blank, 3 point 4 times 10 to the power 11, blank, blank, blank, blank, blank, 4 times 10 to the power 11.

For scientific notation, the "$\times$" symbol is the standard way to show multiplication instead of the dot symbol. Some other examples of scientific notation are $1.2 \times 10^{\text-8}$, $9.99 \times 10^{16}$, and $7 \times 10^{12}$.

Multiplying numbers in scientific notation extends what we do when we multiply regular decimal numbers. For example, one way to find $(80)(60)$ is to view 80 as 8 tens and to view 60 as 6 tens. The product $(80)(60)$ is 48 hundreds or 4,800. Using scientific notation, we can write this calculation as $\displaystyle (8 \times 10^1) (6 \times 10^1) = 48 \times 10^2$ 

To express the product in scientific notation, we would rewrite it as $4.8 \times 10^3$.

Calculating using scientific notation is especially useful when dealing with very large or very small numbers. For example, there are about 39 million, or $3.9 \times 10^7$ residents in California. The state has a water consumption goal of 42 gallons of water per person each day. To find how many gallons of water California would need each day if they met their goal, we can find the product $(42) (3.9 \times 10^7) = 163.8 \times 10^7$, which is equal to $1.638 \times 10^9$. That’s more than 1 billion gallons of water each day.

Comparing very large or very small numbers by estimation also becomes easier with scientific notation. For example, how many ants are there for every human? There are $5 \times 10^{16}$ ants and $8 \times 10^9$ humans. To find the number of ants per human, look at $\frac{5 \times 10^{16}}{8\times 10^9}$. Rewriting the numerator to have the number 50 instead of 5, we get $\frac{50 \times 10^{15}}{8 \times 10^9}$. This gives us $\frac{50}{8} \times 10^6$. Since $\frac{50}{8}$ is roughly equal to 6, there are about $6 \times 10^6$ or 6 million ants per person!