Today, we expand our exponent vocabulary from non-negative exponents to negative exponents.
Negative Exponents
Negative exponents tell us that there are factors downstairs. That is, $$x^{-n}=\frac{1}{x^n}$$ tells us that there are $n$ factors of $x$ in the denominator.
Example: $x^{-1}$
$\displaystyle x^{-1}=\frac{1}{x^1}=\frac{1}{x}$
Example: $x^{-3}$
$\displaystyle x^{-3}=\frac{1}{x^3}$
Example: $\displaystyle \left(\frac{1}{x}\right)^{-1}$
$\displaystyle \left(\frac{1}{x}\right)^{-1}=\frac{1}{\frac{1}{x}}=1\div\frac{1}{x}=1\cdot \frac{x}{1}=1\cdot x =x$
Example: $\displaystyle \left(\frac{1}{x}\right)^{-3}$
$\displaystyle \left(\frac{1}{x}\right)^{-3}
=\frac{1}{\left(\frac{1}{x}\right)^3}
=\frac{1}{\frac{1^3}{x^3}}
=\frac{1}{\frac{1}{x^3}}
=1\div\frac{1}{x^3}
=1\cdot \frac{x^3}{1}
=1\cdot x^3
=x^3$
Example: $\displaystyle \frac{1}{x^{-1}}$
$\displaystyle \frac{1}{x^{-1}}=\frac{1}{\frac{1}{x}}=1\div\frac{1}{x}=1\cdot \frac{x}{1}=1\cdot x =x$
Example: $\displaystyle \frac{1}{x^{-3}}$
$\displaystyle \frac{1}{x^{-3}}=\frac{1}{\frac{1}{x^3}}=1\div\frac{1}{x^3}=1\cdot \frac{x^3}{1}=1\cdot x^3 =x^3$
Example: $2^{-1}+3^{-1}$
$\displaystyle 2^{-1}+3^{-1}=\frac{1}{2}+\frac{1}{3}=\frac{3}{6}+\frac{2}{6}=\frac{5}{6}$
Some General Rules for Negative Exponents
Upstairs/Downstairs 1: $\displaystyle \frac{1}{x^{-n}}=x^{n}$
Upstairs/Downstairs 2: $\displaystyle x^{-n}=\frac{1}{x^n}$ (This is our definition of negative exponent.)
Fraction Flip: $\displaystyle \left(\frac{x}{y}\right)^{-n}=\left(\frac{y}{x}\right)^{n}$
Example: Write the expression without negative exponents. $\displaystyle \frac{y^2}{zx^{-3}}$
$\displaystyle \frac{y^2}{zx^{-3}}=\frac{y^2x^{3}}{z}$
The Quotient Rule: $\displaystyle \frac{x^m}{x^n}=x^{m-n}$
Example: $\displaystyle \frac{x^5}{x^2}$
The long way: $\displaystyle \frac{x^5}{x^2}=\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}=\frac{x \cdot x \cdot x}{1}=x \cdot x \cdot x=x^3$
The shortcut way: $\displaystyle \frac{x^5}{x^2}=x^{5-2}=x^3$
The shortcut way: $\displaystyle \frac{x^5}{x^2}=x^{5-2}=x^3$
Example: $\displaystyle \frac{x^2}{x^5}$
The long way: $\displaystyle \frac{x^2}{x^5}=\frac{x \cdot x}{x \cdot x \cdot x \cdot x \cdot x}=\frac{1}{x \cdot x \cdot x}=\frac{1}{x^3}=x^{-3}$
The shortcut way: $\displaystyle \frac{x^2}{x^5}=x^{2-5}=x^{-3}$
The shortcut way: $\displaystyle \frac{x^2}{x^5}=x^{2-5}=x^{-3}$
Rules for Negative Exponents
The Product Rule: $x^m \cdot x^n=x^{m+n}$
The Quotient Rule: $\displaystyle \frac{x^m}{x^n}=x^{m-n}$
The Power Rule: $(x^m)^n=x^{m \cdot n}$
Power of a Product: $(xy)^n=x^n y^n$
Power of a Quotient: $\displaystyle \left(\frac{x}{y}\right)^n=\frac{x^n}{y^n}$
Example
Simplify the expression by writing it without negative exponents. $$\left(\frac{2y^2}{zx^{-3}}\right)^{-4}$$
Solution #1:
$$
\begin{array}{lll}
\displaystyle \left(\frac{2y^2}{zx^{-3}}\right)^{-4}&\displaystyle=\left(\frac{zx^{-3}}{2y^2}\right)^{4} &\mbox{}\\
\displaystyle &\displaystyle=\left(\frac{z}{2y^2x^{3}}\right)^{4} &\mbox{Upstairs/Downstairs}\\
\displaystyle &\displaystyle=\frac{z^{4}}{(2y^2x^{3})^{4}} &\mbox{A power of a quotient is the quotient of the powers!}\\
\displaystyle &\displaystyle=\frac{z^{4}}{2^{4}(y^2)^{4}(x^{3})^{4}} &\mbox{A power of a product is the product of the powers!}\\
\displaystyle &\displaystyle=\frac{z^{4}}{16y^{2\cdot 4}x^{3\cdot 4}} &\mbox{Power Rule}\\
\displaystyle &\displaystyle=\frac{z^{4}}{16y^{8}x^{12}} &\mbox{}\\
\displaystyle &\displaystyle=\frac{z^{4}}{16x^{12}y^{8}} &\mbox{Alphabetical order is sometimes preferred.}\\
\end{array}
$$
Solution #2:
$$
\begin{array}{lll}
\displaystyle \left(\frac{2y^2}{zx^{-3}}\right)^{-4}&\displaystyle=\frac{(2y^2)^{-4}}{(zx^{-3})^{-4}} &\mbox{A power of a quotient is the quotient of the powers!}\\
\displaystyle &\displaystyle=\frac{2^{-4}(y^2)^{-4}}{z^{-4}(x^{-3})^{-4}} &\mbox{A power of a product is the product of the powers!}\\
\displaystyle &\displaystyle=\frac{2^{-4}y^{2\cdot (-4)}}{z^{-4}x^{(-3)(-4)}} &\mbox{Power Rule}\\
\displaystyle &\displaystyle=\frac{2^{-4}y^{-8}}{z^{-4}x^{12}} &\mbox{simplify exponents}\\
\displaystyle &\displaystyle=\frac{z^{4}}{2^{4}x^{12}y^{8}} &\mbox{Upstairs/Downstairs}\\
\displaystyle &\displaystyle=\frac{z^{4}}{16x^{12}y^{8}} &\mbox{}\\
\end{array}
$$
More Examples
Simplify the following expressions by writing them without negative exponents.
$\displaystyle \left(\frac{t^{7}}{c^{17}}\right)^{-11}$
Solution #1:
$$
\begin{array}{lll}
\displaystyle \left(\frac{t^{7}}{c^{17}}\right)^{-11}&\displaystyle=\left(\frac{c^{17}}{t^{7}}\right)^{11} &\mbox{Fraction Flip}\\
\displaystyle &\displaystyle= \frac{(c^{17})^{11}}{(t^{7})^{11}} &\mbox{A power of a quotient is the quotient of the powers!}\\
\displaystyle &\displaystyle= \frac{c^{187}}{t^{77}} &\mbox{Power Rule}\\
\end{array}
$$
Solution #2:
$$
\begin{array}{lll}
\displaystyle \left(\frac{t^{7}}{c^{17}}\right)^{-11}&\displaystyle=\frac{(t^{7})^{-11}}{(c^{17})^{-11}} &\mbox{A power of a quotient is the quotient of the powers!}\\
\displaystyle &\displaystyle=\frac{t^{7(-11)}}{c^{17(-11)}}&\mbox{Power Rule}\\
\displaystyle &\displaystyle=\frac{t^{-77}}{c^{-187}}&\mbox{}\\
\displaystyle &\displaystyle= \frac{c^{187}}{t^{77}} &\mbox{Upstairs/Downstairs}\\
\end{array}
$$
$\displaystyle \frac{16p^{4}q^{3}}{4p^{5}q^{2}}$
The Exponent Way
$$
\begin{array}{lll}
\displaystyle \frac{16p^{4}q^{3}}{4p^{5}q^{2}}&\displaystyle=\frac{2^4p^{4}q^{3}}{2^2p^{5}q^{2}} &\mbox{}\\
\displaystyle &\displaystyle=2^{4-2}p^{4-5}q^{3-2} &\mbox{Quotient Rule}\\
\displaystyle &\displaystyle=2^{2}p^{-1}q^{1} &\mbox{}\\
\displaystyle &\displaystyle=4p^{-1}q &\mbox{}\\
\displaystyle &\displaystyle=\frac{4q}{p^{1}} &\mbox{Upstairs/Downstairs}\\
\displaystyle &\displaystyle=\frac{4q}{p} &\mbox{}\\
\end{array}
$$
The Traditional Way
$$
\begin{array}{lll}
\displaystyle \frac{16p^{4}q^{3}}{4p^{5}q^{2}}&\displaystyle=\frac{2\cdot 2\cdot 2 \cdot 2 \cdot p \cdot p \cdot p \cdot p \cdot q \cdot q \cdot q}{2 \cdot 2 \cdot p \cdot p \cdot p \cdot p \cdot p \cdot q \cdot q} &\mbox{}\\
\displaystyle &\displaystyle=\frac{2\cdot 2\cdot q}{p} &\mbox{after cancelling}\\
\displaystyle &\displaystyle=\frac{4\cdot q}{p} &\mbox{}\\
\end{array}
$$
$\displaystyle (y^{3} \cdot t^{-7})^{6}(y^{-4} \cdot t^{-2})^{5}$
$$
\begin{array}{lll}
\displaystyle (y^{3} \cdot t^{-7})^{6}(y^{-4} \cdot t^{-2})^{5} &\displaystyle=(y^{3})^{6} \cdot (t^{-7})^{6}(y^{-4})^{5} \cdot (t^{-2})^{5} &\mbox{A power of a product is the product of the powers!}\\
\displaystyle &\displaystyle=y^{3\cdot 6} \cdot t^{-7\cdot 6}\cdot y^{-4\cdot 5}\cdot t^{-2\cdot 5} &\mbox{Power Rule}\\
\displaystyle &\displaystyle=y^{18} \cdot t^{-42}\cdot y^{-20}\cdot t^{-10} &\mbox{}\\
\displaystyle &\displaystyle= t^{-42}t^{-10}\cdot y^{-20}\cdot y^{18} &\mbox{Commutative Property}\\
\displaystyle &\displaystyle= t^{-42+(-10)}\cdot y^{-20+18} &\mbox{Product Rule}\\
\displaystyle &\displaystyle= t^{-52}\cdot y^{-2} &\mbox{}\\
\displaystyle &\displaystyle= \frac{1}{t^{52}\cdot y^{2}} &\mbox{Upstairs/Downstairs}\\
\end{array}
$$
$\displaystyle \left[\frac{(p^{-8} \cdot r^{3})^{2}(p^{-6} \cdot r^{-4})^{7}}{(p^{4} \cdot r^{-5})^{-7}}\right]^{-3}$
$$
\begin{array}{lll}
\displaystyle \left[\frac{(p^{-8} \cdot r^{3})^{2}(p^{-6} \cdot r^{-4})^{7}}{(p^{4} \cdot r^{-5})^{-7}}\right]^{-3}&\displaystyle= \left[\frac{(p^{-8})^{2} \cdot (r^{3})^{2}\cdot (p^{-6})^{7} \cdot (r^{-4})^{7}}{(p^{4})^{-7} \cdot (r^{-5})^{-7}}\right]^{-3}&\mbox{A power of a product is the product of the powers!}\\
\displaystyle &\displaystyle= \left[\frac{p^{-8 \cdot 2} \cdot r^{3 \cdot 2}\cdot p^{-6 \cdot 7} \cdot r^{-4\cdot 7}}{p^{4\cdot(-7)} \cdot r^{(-5)\cdot (-7)}}\right]^{-3}&\mbox{Power Rule}\\
\displaystyle &\displaystyle= \left[\frac{p^{-16} \cdot r^{6}\cdot p^{-42} \cdot r^{-28}}{p^{-28} \cdot r^{35}}\right]^{-3}&\mbox{}\\
\displaystyle &\displaystyle= \left[\frac{p^{-16} \cdot p^{-42} \cdot r^{6} \cdot r^{-28}}{p^{-28} \cdot r^{35}}\right]^{-3}&\mbox{Commutative Property}\\
\displaystyle &\displaystyle= \left[\frac{p^{-16+(-42)} \cdot r^{6+(-28)}}{p^{-28} \cdot r^{35}}\right]^{-3}&\mbox{Product Rule}\\
\displaystyle &\displaystyle= \left[\frac{p^{-58} \cdot r^{-22}}{p^{-28} \cdot r^{35}}\right]^{-3}&\mbox{}\\
\displaystyle &\displaystyle= \left[p^{-58-(-28)} \cdot r^{-22-35}\right]^{-3}&\mbox{Quotient Rule}\\
\displaystyle &\displaystyle= \left[p^{-30} \cdot r^{-57}\right]^{-3}&\mbox{}\\
\displaystyle &\displaystyle=(p^{-30})^{-3} \cdot (r^{-57})^{-3}&\mbox{A power of a product is the product of the powers!}\\
\displaystyle &\displaystyle=p^{90} \cdot r^{171}&\mbox{}\\
\end{array}
$$
Part II
Scientific Notation
Scientific Notation
Scientific Notation
In our universe, there are many scales and sizes of things that we study.
Some scales involve what we can see on earth with only our eyes.
But many interesting phenomena occur on vastly different scales, from microorganisms, molecules, and atoms, to planets, solar systems, and galaxies.
Scientific Notation
When we record numerical facts about the very small, or the very large, we sometimes need a new way to write these numbers.
Scientific notation is how we write really small and really big numbers.
Example
As of the writing of this slide, the population of earth is at about $\mbox{7,900,000,000}$ people.
This is a pretty big number, right?
In fact, instead of saying this number as "$\mbox{seven billion, nine hundred million},$" is easier to say...
$$7.9 \mbox{ billion}$$
Example (Continued)
Now we could simply write "$7.9 \mbox{ billion}$" as "$7.9\times \mbox{1,000,000,000},$" but that's still a bit cumbersome.
What's a more compact way to write $\mbox{1,000,000,000}?$
$$10^9$$
So, putting putting it all together, what's a nice way to write $\mbox{7.9 billion}$ using just a number?
$$7.9\times 10^9$$
This is scientific notation!
Another Example
The number of ways to arrange a standard, $52$-card deck is $$52\cdot 51 \cdot 50 \cdot 49 \cdots 4 \cdot 3 \cdot 2 \cdot 1.$$
Using regular notation, this number is equal to...
$$
80658175170943878571660636856403766975289505440883277824000000000000
$$
Another Example (Continued)
We can write $$ 80658175170943878571660636856403766975289505440883277824000000000000 $$ in scientific notation as...
$$8.07 \times 10^{67}$$
We note that this number is only an approximation of the number above, but when it comes to numbers this big,
we're generally interested in the magnitude.
For comparison, the number of atoms in the that make up earth is on the order of $1.33\times 10^{50}.$
How we converted to scientific notation: $$ \begin{array}{lll} \displaystyle &\displaystyle 80658175170943878571660636856403766975289505440883277824000000000000 &\mbox{}\\ \displaystyle \approx &\displaystyle 80700000000000000000000000000000000000000000000000000000000000000000 &\mbox{}\\ \displaystyle = &\displaystyle 8.07\times 10000000000000000000000000000000000000000000000000000000000000000000 &\mbox{}\\ \displaystyle = &\displaystyle 8.07\times 10^{67} &\mbox{}\\ \end{array} $$ We'll get more into this a little later.
For comparison, the number of atoms in the that make up earth is on the order of $1.33\times 10^{50}.$
How we converted to scientific notation: $$ \begin{array}{lll} \displaystyle &\displaystyle 80658175170943878571660636856403766975289505440883277824000000000000 &\mbox{}\\ \displaystyle \approx &\displaystyle 80700000000000000000000000000000000000000000000000000000000000000000 &\mbox{}\\ \displaystyle = &\displaystyle 8.07\times 10000000000000000000000000000000000000000000000000000000000000000000 &\mbox{}\\ \displaystyle = &\displaystyle 8.07\times 10^{67} &\mbox{}\\ \end{array} $$ We'll get more into this a little later.
Really Small Numbers
The distance between the silicon carbide atoms shown below is about $$ 0.0000000003 \mbox{ meters} $$
Let's write this in scientific notation!
$$
\begin{array}{lll}
\displaystyle 0.0000000003&\displaystyle= \frac{3}{\mbox{10,000,000,000}}&\mbox{}\\
\displaystyle &\displaystyle= \frac{3}{10^{10}} &\mbox{}\\
\displaystyle &\displaystyle= 3\times \frac{1}{10^{10}} &\mbox{}\\
\displaystyle &\displaystyle= 3\times 10^{-10} &\mbox{}\\
\end{array}
$$
Really small numbers use negative exponents.
Scientific Notation
A number written in scientific notation has the form $$ a \times 10^{n} $$ where $a$ is a number $1$ or larger, but smaller than $10,$ and $n$ is an integer.
Examples $$ \begin{array}{lll} \displaystyle &\displaystyle 8.07\times 10^{67} &\mbox{the approximate number of arrangements of a 52-card deck}\\ \displaystyle &\displaystyle 1.33\times 10^{50} &\mbox{the approximate number of atoms in the earth}\\ \displaystyle &\displaystyle 3\times 10^{-10} &\mbox{the distance between atoms in a silicon carbide sheet}\\ \end{array} $$
Converting from Decimals to Scientific Notation
1) Move the decimal point so that the first factor is greater than or equal to $1$ but less than $10.$
2) Count the number of decimal places, $n,$ that the decimal point was moved.
3) Write the number as a product with a power of $10.$
If the original number is:
- greater than $1,$ the power of $10$ will be $10^n.$
- between $0$ and $1,$ the power of $10$ will be $10^{-n}.$
More Examples
Convert the numbers to scientific notation.
$\mbox{431000}$
$$
431000=4.31\times 10^{5}
$$
$0.00000075$
$$
0.00000075=7.5\times 10^{-7}
$$
A Fun Question
Approximately many times larger is the number of arrangement in a deck of cards than the number of atoms on earth?
Are you at least a little bit curious to see the answer?
To compare these numbers we divide:
$$
\frac{8.07\times 10^{67}}{1.33\times 10^{50}}
$$
Question: But what do we do now???
Answer: What we've been doing this whole time: arithmetic! $$ \begin{array}{lll} \displaystyle \frac{8.07\times 10^{67}}{1.33\times 10^{50}}&\displaystyle=\frac{8.07}{1.33}\times\frac{10^{67}}{10^{50}} &\mbox{This is fraction multiplication!}\\ \displaystyle &\displaystyle\approx 6.07\times 10^{17}&\mbox{using rules of exponents!}\\ \end{array} $$ That is, there are about $6.07\times 10^{17}=\mbox{607,000,000,000,000,000},$ or six hundred and seven quadrillion times more arrangements in a deck of cards than there are atoms on earth.
Answer: What we've been doing this whole time: arithmetic! $$ \begin{array}{lll} \displaystyle \frac{8.07\times 10^{67}}{1.33\times 10^{50}}&\displaystyle=\frac{8.07}{1.33}\times\frac{10^{67}}{10^{50}} &\mbox{This is fraction multiplication!}\\ \displaystyle &\displaystyle\approx 6.07\times 10^{17}&\mbox{using rules of exponents!}\\ \end{array} $$ That is, there are about $6.07\times 10^{17}=\mbox{607,000,000,000,000,000},$ or six hundred and seven quadrillion times more arrangements in a deck of cards than there are atoms on earth.
Other Examples of Arithmetic in Scientific Notation
Multiply: $(4.31 \times 10^5)\cdot (7.5\times 10^{-7})$
$$
\begin{array}{lll}
\displaystyle (4.31 \times 10^5)\cdot (7.5\times 10^{-7})&\displaystyle=(4.31 \times 10^5)\cdot (7.5\times 10^{-7}) &\mbox{commutative property}\\
\displaystyle &\displaystyle=32.325\times 10^{-2} &\mbox{exponent rules}\\
\displaystyle &\displaystyle=32.325\times 10^{-2} &\mbox{not in scientific notation yet}\\
\displaystyle &\displaystyle=3.2325 \times 10^{1} \times 10^{-2} &\mbox{convert 32.325 to scientific notation}\\
\displaystyle &\displaystyle=3.2325 \times 10^{-1} &\mbox{}\\
\end{array}
$$
Divide: $(4.31 \times 10^5)\div (7.5\times 10^{-7})$
$$
\begin{array}{lll}
\displaystyle (4.31 \times 10^5)\div (7.5\times 10^{-7})&\displaystyle=\frac{4.31 \times 10^5}{7.5\times 10^{-7}} &\mbox{}\\
\displaystyle &\displaystyle= \frac{4.31}{7.5} \times \frac{10^5}{10^{-7}} &\mbox{}\\
\displaystyle &\displaystyle\approx 0.575 \times 10^{12} &\mbox{not in scientific notation yet}\\
\displaystyle &\displaystyle= 5.75\times 10^{-1} \times 10^{12} &\mbox{convert 0.575 to scientific notation}\\
\displaystyle &\displaystyle= 5.75\times 10^{11} &\mbox{}\\
\end{array}
$$
Converting Back from Scientific Notation to Decimals
1) Determine the exponent, $n.$ on the factor $10.$
2) Move the decimal $n$ places, adding zeros if needed.
- If the exponent is positive, move the decimal point $n$ places to the right
- If the exponent is negative, move the decimal point $|n|$ places to the left.
Examples
Convert from scientific to decimal notation.
$9.25 \times 10^4.$
$$9.25 \times 10^4=92500$$
$1.3 \times 10^{-3}.$
$$1.3 \times 10^{-3}=0.0013$$
The Scales in Our Universe