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6.1: Exponents

Created by: CK-12

Name: ________________________

Rules of Exponents

1. Here are the first six powers of two.

$2^1 & = 2\\2^2 & = 4\\2^3 & = 8\\2^4 & = 16\\2^5 & = 32\\2^6 & = 64$

a. If I asked you for $2^7$ (without a calculator), how would you get it? More generally, how do you always get from one term in this list to the next term? ________________

b. Write an algebraic generalization to represent this rule. _____________________

2. Suppose I want to multiply $2^5$ times $2^3$. Well, $2^5$ means $2 \times 2 \times 2 \times 2 \times 2$, and $2^3$ means $2 \times 2 \times 2$. So we can write the whole thing out like this.

$& \qquad \quad 2^5 \qquad \qquad \qquad \quad 2^3 \qquad = \qquad \qquad \qquad 2^8\\& \overbrace{2*2*2*2*2 } \quad * \quad \overbrace{ 2*2*2 } \quad = \quad \overbrace{ 2*2*2*2*2*2*2*2}$

a. This shows that $(2^5)(2^3)= 2^{[ \ ]}$

b. Using a similar drawing, demonstrate what $(10^3)(10^4)$ must be.

c. Now, write an algebraic generalization for this rule. ____________________

d. Show how your answer to $1b$ (the “getting from one power of two, to the next in line”) is a special case of the more general rule you came up with in $2c$ (“multiplying two exponents”).

3. Now we turn our attention to division. What is $\frac{3^{12}}{3^{10}}$?

a. Write it out explicitly. (Like earlier I wrote out explicitly what $2^52^3$ was: expand the exponents into a big long fraction.)

b. Now, cancel all the like terms on the top and the bottom. (That is, divide the top and bottom by all the $3s$ they have in common.)

c. What you are left with is the answer. So fill this in: $\frac{3^{12}}{3^{10}} = 3^{[ \ ]}$.

d. Write a generalization that represents this rule.

e. Suppose we turn it upside-down. Now, we end up with some $3s$ on the bottom. Write it out explicitly and cancel $3s$, as you did before:

$\frac{3^{10}}{3^{12}} = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;} = \frac{1}{3^{[ \ ]}}$

f. Write a generalization for the rule in part (e). Be sure to mention when that generalization applies, as opposed to the one in part (d)!

4. Use all those generalizations to simplify $\frac{x^3y^3x^7}{x^5y^5}$

5. Now we’re going to raise exponents, to exponents. What is $(2^3)^4?$ Well, $2^3$ means $2 \times 2 \times 2$. And when you raise anything to the fourth power, you multiply it by itself, four times. So we’ll multiply that by itself four times:

$(2^3)^4 = (2 \times 2 \times 2) \ (2 \times 2 \times 2) \ (2 \times 2 \times 2) \ (2 \times 2 \times 2)$

a. So, just counting $2s, \ (2^3)^4 = 2^{[ \ ]}$.

b. Expand out $(10^5)^3$ in a similar way, and show what power of $10$ it equals.

c. Find the algebraic generalization that represents this rule.

Name: ________________________

Homework: Rules of Exponents

Memorize these:

$x^ax^b & = x^{a+b}\\\frac{x^a}{x^b}& = x^{a-b} \ \text{or} \ \frac{1}{x^{b-a}}\\(x^a)^b & = x^{ab}$

Simplify, using these rules

1. $3^{10} \times 3^5$
2. $\frac{3^{10}}{3^5}$
3. $\frac{3^5}{3^{10}}$
4. $(3^5)^{10}$
5. $(3^{10})^5$
6. $3^{10} + 3^5$
7. $3^{10} - 3^5$
8. $\frac{6x^3y^2z^5}{4x^{10}yz^2}$
9. $\frac{6x^2y^3+3x^3y}{xy+x^2y}$
10. $\frac{(3x^2y)^2+xy}{(xy)^3}$
11. $(3x^2 + 4xy)^2$

Name: ______________________

Extending the Idea of Exponents

1. Complete the following table.

$x$ $3^x$
$4$ $3 \times 3 \times 3 \times 3 = 81$
$3$
$2$
$1$

2. In this table, every time you go to the next row, what happens to the left-hand number $(x)$?

3. What happens to the right-hand number $(3^x)$?

4. Now, let’s assume that pattern continues, and fill in the next few rows.

$x$ $3^x$
$1$

5. Based on this table, $3^0 =$

6. $3^{-1} =$

7. $3^{-2} =$

8. What would you expect $3^{-4}$ to be?

9. Now check $3^{-4}$ on your calculator. Did it come out the way you predicted?

Name: ______________________

Homework: Extending the Idea of Exponents

1. $5^0 =$

2. $5^{-2} =$

3. $(-2)^{-2} =$

4. $(-2)^{-3} =$

5. $6^{-3} =$

6. $x^0 =$

7. $x^{-a} =$

In those last two problems, of course, you have created the general rules for zero and negative exponents. So hey, what happens to our trusty rules of exponents? Let’s try...

8. Let’s look at the problem $6^06^x$ two different ways.

a. What is $6^0?$ Based on that, what is $6^06^x?$

b. What do our rules of exponents tell us about $6^06^x?$

9. Let’s look at the problem $\frac{6^0}{6^x}$ two different ways.

a. What is $6^0?$ Based on that, what is $\frac{6^0}{6^x}?$

b. What do our rules of exponents tell us about $\frac{6^0}{6^x}?$

10. Let’s look at the problem $6^{-4}6^3$ two different ways.

a. What does $6^{-4}$ mean? Based on that, what is $6^{-4}6^3?$

b. What do our rules of exponents tell us about $6^{-4}6^3?$

11. What would you square if you wanted to get $x^{36}?$

Now let’s solve a few equations.

12. Solve for $x: \ 3^{x+2} = 3^{8-x}$. (Hint: If the bases are the same, the exponents must be the same!)

13. Solve for $x: \ 2^{4x-3} = 8^{x-2}$. (Hint: Start by rewriting $8$ as $2^3$, then use the rules of exponents.)

14. Solve for $x: \ 5^{(3x^2+13x+10)} = 25^{(x+2)}$. (No more hints this time, you’re on your own.)

15. Solve for $x: \ (7^x)(7^{(x+2)}) = 1$

16. Solve for $x: \ (7^x)^{(x+2)} = 1$

Name: __________________

Fractional Exponents

1. On the homework, we demonstrated the rule of negative exponents by building a table. Now, we’re going to demonstrate it another way—by using the rules of exponents.

a. According to the rules of exponents, $\frac{7^3}{7^5}= 7^{[ \ ]}$.

b. But if you write it out and cancel the excess $7s$, then $\frac{7^3}{7^5}= \underline{\;\;\;\;\;\;\;\;}$.

c. Therefore, since $\frac{7^3}{7^5}$ can only be one thing, we conclude that these two things must be equal: write that equation!

2. Now, we’re going to approach fractional exponents the same way. Based on our rules of exponents, $(9^{\frac{1}{2}})^2 = \underline{\;\;\;\;\;\;\;\;}$

3. So, what does that tell us about $9^{\frac{1}{2}}?$ Well, it is some number that when you square it, you get _______ (* same answer you gave for number $2$). So therefore, $9^{\frac{1}{2}}$ itself must be: _______

4. Using the same logic, what is $16^{\frac{1}{2}}?$

5. What is $25^{\frac{1}{2}}?$

6. What is $x^{\frac{1}{2}}?$

7. Construct a similar argument to show that $8^{\frac{1}{3}}? = 2$.

8. What is $27^{\frac{1}{3}}?$

9. What is $(-1)^{\frac{1}{3}}?$

10. What is $x^{\frac{1}{3}}?$

11. What would you expect $x^{\frac{1}{5}}$ to be?

12. What is $25^{-\frac{1}{2}}?$ (You have to combine the rules for negative and fractional exponents here!)

13. Check your answer to $^\#12$ on your calculator. Did it come out the way you expected?

OK, we’ve done negative exponents, and fractional exponents—but always with a $1$ in the numerator. What if the numerator is not $1?$

14. Using the rules of exponents, $\left (8^{\frac{1}{3}}\right )^2 = 8^{[ \ ]}$.

So that gives us a rule! We know what $\left (8^{\frac{1}{3}}\right )^2$ is, so now we know what $8^{\frac{2}{3}}$ is.

15. $8^{\frac{2}{3}} =$

16. Construct a similar argument to show what $16^{\frac{3}{4}}$ should be.

17. Check $16^{\frac{3}{4}}$ on your calculator. Did it come out the way you predicted?

Now let’s combine all our rules! For each of the following, say what it means and then say what actual number it is. (For instance, for $9^{\frac{1}{2}}$ you would say it means $\sqrt{9}$ so it is $3$.)

18. $8^{-\frac{1}{3}} =$

19. $8^{-\frac{2}{3}} =$

For these problems, just say what it means. (For instance, $3^{\frac{1}{2}}$ means $\sqrt{3}$, end of story.)

20. $10^{-4}$

21. $2^{-\frac{3}{4}}$

22. $x^{\frac{a}{b}}$

23. $x^{-\frac{a}{b}}$

Name: __________________

Homework: Fractional Exponents

We have come up with the following definitions.

$x^0 & = 1\\x^{-a} & = \frac{1}{x^a}\\x^{\frac{a}{b}} & = \sqrt[b]{x^a}$

Let’s get a bit of practice using these definitions.

1. $100^{\frac{1}{2}}=$

2. $100^{-2} =$

3. $100^{-\frac{1}{2}} =$

4. $100^{\frac{3}{2}} =$

5. $100^{-\frac{3}{2}} =$

6. Check all of your answers above on your calculator. If any of them did not come out right, figure out what went wrong, and fix it!

7. Solve for $x: \frac{x^{\frac{3}{2}}}{x^{\frac{1}{2}}} = 17^{\frac{1}{2}}$

8. Solve for $x: x^{\frac{1}{2}} = 9$

9. Simplify: $\frac{x}{\sqrt{x}}$

10. Simplify: $\frac{x^{\frac{3}{2}}+\sqrt{x}}{x^{\frac{5}{2}}+\frac{1}{\sqrt{x}}}$ (Hint: Multiply the top and bottom by $x^{\frac{1}{2}}$.)

Now...remember inverse functions? You find them by switching the $x$ and the $y$ and then solving for $y$. Find the inverse of each of the following functions. To do this, in some cases, you will have to rewrite the things. For instance, in $^{\#}9$, you will start by writing $y=x^{\frac{1}{2}}$. Switch the $x$ and the $y$, and you get $x=y^{\frac{1}{2}}$. Now what? Well, remember what that means: it means $x=\sqrt{y}$. Once you’ve done that, you can solve for $y$, right?

11. $x^3$

a. Find the inverse function.

b. Test it.

12. $x^{-2}$

a. Find the inverse function.

b. Test it.

13. $x^0$

a. Find the inverse function.

b. Test it.

14. Can you find a generalization about the inverse function of an exponent?

15. Graph $y=2^x$ by plotting points. Make sure to include both positive and negative $x$ values.

16. Graph $y=2 \times 2^x$ by doubling all the $y-$values in the graph of $y=2^x$.

17. Graph $y=2^{x+1}$ by taking the graph $y=2^x$ and “shifting” it to the left by one.

18. Graph $y=\left (\frac{1}{2}\right )^x$ by plotting points. Make sure to include both positive and negative $x$ values.

Name: ___________________________

“Real Life” Exponential Curves

The Famous King Exponent Story

This is a famous ancient story that I am not making up, except that I am changing some of the details.

A man did a great service for the king. The king offered to reward the man every day for a month. So the man said: “Your Majesty, on the first day, I want only a penny. On the second day, I want twice that: $2$ . On the third day, I want twice that much again: $4$ pennies. On the fourth day, I want $8$ pennies, and so on. On the thirtieth day, you will give me the last sum of money, and I will consider the debt paid off.”

The king thought he was getting a great deal...but was he? Before you do the math, take a guess: how much do you think the king will pay the man on the $30^{\mathrm{th}}$ day? _______________

Now, let’s do the math. For each day, indicate how much money (in pennies) the king paid the man. Do this without a calculator, it’s good practice and should be quick.

Day 1: $\underline{1\;\mathrm{penny}}$

Day 2: $\underline{\;\;\;\;\;2\;\;\;\;\;}$

Day 3: $\underline{\;\;\;\;\;4\;\;\;\;\;}$

Day 4: $\underline{\;\;\;\;\;8\;\;\;\;\;}$

Day 5: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 6: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 7: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 8: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 9: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 10: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 11: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 12: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 13: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 14: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 15: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 16: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 17: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 18: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 19: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 20: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 21: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 22: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 23: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 24: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 25: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 26: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 27: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 28: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 29: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Day 30: $\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Now let’s get mathematical. On the $n^{\mathrm{th}}$ day, how many pennies did the king give the man? ______

Use your calculator, and the formula you just wrote down, to answer the question: what did the king pay the man on the $30^{\mathrm{th}}$ day? _______ Does it match what you put under “Day $30$” above? (If not, something’s wrong somewhere—find it and fix it!)

Finally, do a graph of this function, where the “day” is on the $x-$axis and the “pennies” is on the $y$ axis (so you are graphing pennies as a function of day). Obviously, your graph won’t get past the fifth or sixth day or so, but try to get an idea for what the shape looks like.

Compound Interest

Here is a slightly more realistic situation. Your bank pays $6\%$ interest, compounded annually.That means that after the first year, they add $6\%$ to your money. After the second year, they add another $6\%$ to the new total...and so on.

You start with $\1,000$. Fill in the following table.

Year The bank gives you this... and you end up with this
$0$ $0$ $\1000$
$1$ $\60$ $\1060$
$2$ $\63.60$
$3$
$4$
$5$

Now, let’s start generalizing. Suppose at the end of one year, you have $x$ dollars. How much does the bank give you that year?

And when you add that, how much do you have at the end of the next year? (Simplify as much as possible.)

So, now you know what is happening to your money each year. So after year n, how much money do you have? Give me an equation.

Test that equation to see if it gives you the same result you gave above for the end of year $5$.

Once again, graph that. The $x-$axis should be year. The $y-$axis should be the total amount of money you end up with after each year.

How is this graph like, and how is it unlike, the previous graph?

If you withdraw all your money after $\frac{1}{2}$ a year, how much money will the bank give you? (Use the equation you found above!)

If you withdraw all your money after $2^{\frac{1}{2}}$ years, how much money will the bank give you?

Suppose that, instead of starting with $\1,000$, I just tell you that you had $\1,000$ at year $0$. How much money did you have five years before that (year $-5$)?

How many years will it take for your money to triple? That is to say, in what year will you have $\3,000?$

Name: ___________________________

Homework: “Real Life” Exponential Curves

Radioactive substances decay according to a “half-life.” The half-life is the period of time that it takes for half the substance to decay. For instance, if the half-life is $20\;\mathrm{minutes}$, then every $20\;\mathrm{minutes}$, half the remaining substance decays.

As you can see, this is the sort of exponential curve that goes down instead of up: at each step (or half-life) the total amount divides by $2$; or, to put it another way, multiplies by $\frac{1}{2}$.

You have $1\;\mathrm{gram}$ of a substance with a half-life of $1\;\mathrm{minute}$. Fill in the following table.

Time Substance remaining
$0$ $1\;\mathrm{gram}$
$1\;\mathrm{minute}$ $\frac{1}{2}\;\mathrm{gram}$
$2\;\mathrm{minutes}$
$3\;\mathrm{minutes}$
$4\;\mathrm{minutes}$
$5\;\mathrm{minutes}$

After $n$ minutes, how many grams are there? Give me an equation.

Use that equation to answer the question: after $5\;\mathrm{minutes}$, how many grams of substance are there? Does your answer agree with what you put under “$5\;\mathrm{minutes}$” above? (If not, something’s wrong somewhere—find it and fix it!)

How much substance will be left after $4\frac{1}{2}\;\mathrm{minutes}?$

How much substance will be left after half an hour?

How long will it be before only one one-millionth of a gram remains?

Finally, on the attached graph paper, do a graph of this function, where the “minute” is on the $x-$axis and the “amount of stuff left” is on the $y-$axis (so you are graphing grams as a function of minutes). Obviously, your graph won’t get past the fifth or sixth minute or so, but try to get an idea for what the shape looks like.

Now, we’re going to do a more complicated example. Let’s say you start with $1000\;\mathrm{grams}$ of a substance, and its half-life is $20\;\mathrm{minutes}$; that is, every $20\;\mathrm{minutes}$, half the substance disappears. Fill in the following chart.

Time Half-Lives Substance remaining
$0$ $0$ $1000\;\mathrm{grams}$
$20\;\mathrm{minutes}$ $1$ $500\;\mathrm{grams}$
$40\;\mathrm{minutes}$
$60\;\mathrm{minutes}$
$80\;\mathrm{minutes}$
$100\;\mathrm{minutes}$

After $n$ half-lives, how many grams are there? Give me an equation.

After $n$ half-lives, how many minutes have gone by? Give me an equation.

Now, let’s look at that equation the other way. After $t$ minutes (for instance, after $60 \;\mathrm{minutes}$, or $80\;\mathrm{minutes}$, etc), how many half-lives have gone by? Give me an equation.

Now we need to put it all together. After $t$ minutes, how many grams are there? This equation should take you directly from the first column to the third: for instance, it should turn $0$ into $1000$, and $20$ into $500$. (*Note: you can build this as a composite function, starting from two of your previous answers!)

Test that equation to see if it gives you the same result you gave above after $100 \;\mathrm{minutes}$.

Once again, graph that on graph paper. The $x-$axis should be minutes. The $y-$axis should be the total amount of substance. In the space below, answer the question: how is it like, and how is it unlike, the previous graph?

How much substance will be left after $70\;\mathrm{minutes}$?

How much substance will be left after two hours? (*Not two minutes, two hours!)

How long will it be before only one gram of the original substance remains?

Finally, a bit more about compound interest

If you invest $^\A$ into a bank with $i\%$ interest compounded $n$ times per year, after $t$ years your bank account is worth an amount $M$ given by:

$M=A \left (1+\frac{i}{n}\right )^{\mathrm{nt}}$

For instance, suppose you invest $^\1,000$ in a bank that gives $10\%$ interest, compounded “semi-annually” (twice a year). So $A$, your initial investment, is $^\1,000$. $i$, the interest rate, is $10\%$, or $0.10$. $n$, the number of times compounded per year, is $2$. So after $30$ years, you would have:

$^\1,000 \left (1+\frac{0.10}{2}\right )^{2 \times 30} =^\18,679$. (Not bad for a $^\1,000$ investment!)

Now, suppose you invest $^\1.00$ in a bank that gives $100\%$ interest (nice bank!). How much do you have after one year if the interest is...

• Compounded annually (once per year)?
• Compounded quarterly (four times per year)?
• Compounded daily?
• Compounded every second?

Name: ______________

Sample Test: Exponents

1. $x^{-6}$

2. $x^0$

3. $x^{\frac{1}{8}}$

4. $x^{\frac{2}{3}}$

5. $\left ({\frac{2}{3}}\right )^2$

6. $\left ({\frac{1}{2}}\right )^{-x}$

7. $(-2)^2$

8. $(-2)^3$

9. $(-2)^{-1}$

10. $(-9)^{\frac{1}{2}}$

11. $(-8)^{\frac{1}{3}}$

12. $y^{\frac{1}{4}}y^{\frac{3}{4}}$

13. $\frac{4x^4y^5z}{6wxy^2z^3}$

14. $\left (x^{\frac{1}{3}}\right )^3$

15. $\left (x^{\frac{1}{3}}\right )^2$

16. $x^{\frac{6}{3}}$

17. $x^{-\frac{3}{4}}$

18. $(4 \times 9)^{\frac{1}{2}}$

19. $4^{\frac{1}{2}} \times 9^{\frac{1}{2}}$

20. Give an algebraic formula that gives the generalization for $^\#18-19$.

Solve for $x$.

21. $8^x = 64$

22. $8^x = 8$

23. $8^x = 1$

24. $8^x = 2$

25. $8^x = \frac{1}{8}$

26. $8^x = \frac{1}{64}$

27. $8^x = \frac{1}{2}$

28. $8^x = 0$

29. Rewrite $\frac{1}{\sqrt[3]{x^2}}$ as $X^\mathrm{something}$.

Solve for $x$.

30. $2^{(x+3)}2^{(x+4)} = 2$

31. $3^{(x^2)} = \left (\frac{1}{9}\right )^{3x}$

32. A friend of yours is arguing that $x^{\frac{1}{3}}$ should be defined to mean something to do with “fractions, or division, or something.” You say, “No, it means _____ instead.” He says, “That’s a crazy definition!” Give him a convincing argument why it should mean what you said it means.

33. On October $1^{\mathrm{st}}$, I place $3$ sheets of paper on the ground. Each day thereafter, I count the number of sheets on the ground, and add that many again. (So if there are $5$ sheets, I add $5$ more.) After I add my last pile on Halloween (October $31^\mathrm{st}$), how many sheets are there total?

a. Give me the answer as a formula.

b. Plug that formula into your calculator to get a number.

c. If one sheet of paper is $\frac{1}{250}$ inches thick, how thick is the final pile?

34. Depreciation. The Web site www.bankrate.com defines depreciation as “the decline in a car’s value over the course of its useful life” (and also as “something new-car buyers dread”). The site goes on to say:

Let’s start with some basics. Here’s a standard rule of thumb about used cars. A car loses $15\;\mathrm{percent}$ to $20\;\mathrm{percent}$ of its value each year.

For the purposes of this problem, let’s suppose you buy a new car for exactly $^\10,000$, and it loses only $15\%$ of its value every year.

a. How much is your car worth after the first year?

b. How much is your car worth after the second year?

c. How much is your car worth after the $n^{\mathrm{th}}$ year?

d. How much is your car worth after ten years? (This helps you understand why new-car buyers dread depreciation.)

35. Draw a graph of $y=2 \times 3^x$. Make sure to include negative and positive values of $x$.

36. Draw a graph of $y=\left (\frac{1}{3}\right )^x-3$. Make sure to include negative and positive values of $x$.

37. What are the domain and range of the function you graphed in number $36?$

Extra credit: We know that $(a+b)^2$ is not, in general, the same as $a^2+b^2$. But under what circumstances, if any, are they the same?

Feb 23, 2012

Apr 29, 2014