Base, exponent and the value of a power
When the same number is multiplied by itself again and again, how do you write it short, and how do you work out what it comes to?
Where you are 3 in Exponent notation
Why this exists
Tali hears some news and phones three friends. Each of them phones three new people, and each of those phones three more, and so on. Count the calls round by round. Round one has 3. Round two has 3 \times 3, because each of the first three makes 3 calls. Round three has 3 \times 3 \times 3. By round twenty the product has twenty threes in it, a line long enough to lose count in.
Folding a sheet of paper in half does the same thing with twos. Each fold doubles the number of layers, so after twenty folds the count of layers is twenty twos multiplied together.
Multiplying the same number by itself again and again turns up so often that it has its own short way of writing: a small raised number that counts the factors. This lesson is about that notation and about the second question that comes straight after it, which is how large the result is. Multiplication facts are all it needs.
Definition
A power is a product in which every factor is the same number. It is written with two numbers:
- the base, the number being multiplied, written at the usual size;
- the exponent, a smaller number raised up beside the base, which counts how many factors of the base are multiplied together.
Here 3 is the base and 4 is the exponent: four factors, each of them 3. The expression 3^4 is called a power of 3, and its value, 81, is also called a power of 3: 81 is the fourth power of 3. Writing 3^4 is exponential form; writing 3 \times 3 \times 3 \times 3 is expanded form.
An exponent of 1 means one factor, so the power is just the base: 19^1 = 19. In this lesson every exponent is a counting number, 1, 2, 3 and so on, because its whole job is to count.
Reading a power aloud. The exponent is said as an ordinal, “to the fourth power”, often shortened to “to the fourth”. The second and third powers have their own names.
| Written | Said |
|---|---|
| 3^4 | three to the fourth power, or three to the fourth |
| 7^8 | seven to the eighth power |
| 20^2 | twenty squared, or twenty to the second power |
| 20^3 | twenty cubed, or twenty to the third power |
The names come from shapes. A square with sides 20 units long covers 20^2 square units, and a cube with edges 20 units long fills 20^3 cubic units. The lists and pictures of those numbers belong to Perfect squares and perfect cubes.
A fraction as the base needs brackets. \left(\tfrac{2}{3}\right)^2 means \tfrac{2}{3} \times \tfrac{2}{3}: the whole fraction is the base. Without the brackets, \tfrac{2^2}{3} puts the exponent on the 2 alone. A decimal written as one string of digits is one number, so 0.3^2 means 0.3 \times 0.3 with no brackets needed.
Intuition
The exponent counts multiplications in the way that a multiplication counts additions. 3 \times 4 adds four threes: 3 + 3 + 3 + 3 = 12. 3^4 multiplies four threes: 3 \times 3 \times 3 \times 3 = 81. Same two numbers, different job, very different size.
Figure 1 shows Tali’s chain. Everyone called in one round makes 3 calls in the next, so each round has 3 times as many calls as the round before it, and round three has three factors of 3.
Set the chain beside the steady climb you get from adding threes:
| Round | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Calls in that round | 3^1 = 3 | 3^2 = 9 | 3^3 = 27 | 3^4 = 81 |
| Threes added | 3 \times 1 = 3 | 3 \times 2 = 6 | 3 \times 3 = 9 | 3 \times 4 = 12 |
Adding grows by the same amount each time. Multiplying grows by the same factor each time, and that runs away fast.
Folding is the same picture with a base of 2. One fold puts one half on top of the other, so the layers double, and each new fold doubles them again.
Results
The exponent counts factors of the base. 3^4 is four factors of 3, which is 81. It is not 3 \times 4 = 12, which is four threes added.
Base and exponent cannot be swapped. 3^4 = 81 but 4^3 = 4 \times 4 \times 4 = 64.
An exponent of 1 gives the base. One factor of a number is the number.
Powers of 1 and of 0. Every power of 1 is 1, because every factor is 1. Every power of 0 is 0, because a product with a factor of 0 is 0.
Powers of ten. 10^1 = 10, 10^2 = 100, 10^3 = 1000: the exponent is the number of zeros after the 1. Each factor of ten moves every digit one place to the left, as in Multiplying and dividing by ten, one hundred and one thousand. The full story, and what these numbers do for place value, is in Powers of ten.
Powers of a fraction. Multiplying fractions multiplies the tops and multiplies the bottoms, so a power of a fraction is the power of the top over the power of the bottom:
Powers of a decimal. Decimal places add when decimals multiply. A number with one decimal place has two decimal places when squared and three when cubed.
Bigger or smaller. When the base is more than 1, each extra factor makes the power larger. When the base is between 0 and 1, each extra factor makes it smaller.
Why it is true
The notation itself is a definition: 3^4 is the name agreed for 3 \times 3 \times 3 \times 3. Everything else follows from counting factors.
Why the chain gives powers. Round one has 3 calls. Each call in a round leads to 3 calls in the next round, so going from one round to the next multiplies the count by 3. After four rounds the count is 3 multiplied together four times, which is 3^4. The same argument with doubling gives the folds: after five folds there are five factors of 2.
Why ones and zeros behave. 1 \times 1 \times 1 \times 1 = 1, however many ones there are. And once a product contains a 0, the whole product is 0.
Why powers of ten are a 1 and some zeros. Start from 1. Multiplying by 10 moves every digit one place left and puts a 0 in the ones place. Each factor of ten does this once, so six factors leave a 1 with six zeros after it.
Why a fraction squared is the tops over the bottoms. Figure 3 shows a square with sides of length 1. Cut it into thirds across and into thirds down, making 3 \times 3 = 9 equal pieces. The shaded rectangle is \tfrac{2}{3} wide and \tfrac{2}{3} tall, and it covers 2 \times 2 = 4 of the pieces. So \tfrac{2}{3} \times \tfrac{2}{3} = \tfrac{4}{9}: the tops multiply to count shaded pieces and the bottoms multiply to count all the pieces.
Why decimal places add. 0.3 = \tfrac{3}{10}, so 0.3 \times 0.3 = \tfrac{3 \times 3}{10 \times 10} = \tfrac{9}{100} = 0.09. Tenths times tenths are hundredths: one place and one place make two.
Why the size goes up or down. Multiplying by a number more than 1 gives more than you started with. Multiplying by a number between 0 and 1 takes only part of what you started with, so it gives less. A power just does this over and over.
Worked examples
The lesson is taught in two parts. Examples 1 to 3 belong to the first: naming the parts of a power, reading it aloud, and moving between a repeated product and a power. Examples 4 to 7 belong to the second: working out the value of a power, with whole numbers, fractions and decimals as the base.
Naming the parts and reading aloud
For each power, name the base and the exponent, write it in expanded form, and say it aloud. (a) 6^2 (b) 4^3 (c) 11^5 (d) 1.5^2
Solution(a) The base is 6, the number at the usual size, and the exponent is 2, the raised number. Two factors of 6: 6 \times 6. Say “six squared”, or “six to the second power”.
(b) The base is 4 and the exponent is 3: 4 \times 4 \times 4. Say “four cubed”, or “four to the third power”.
(c) The base is 11 and the exponent is 5: 11 \times 11 \times 11 \times 11 \times 11. Powers past the third have no special name, so say “eleven to the fifth power”, or “eleven to the fifth”.
(d) The base is all of 1.5, not just the 5 after the point, because the digits are written as one number. The exponent is 2: 1.5 \times 1.5. Say “one point five squared”.
From a repeated product to a power, and back
Write (a) and (b) as powers, and write (c) in expanded form. (a) 9 \times 9 \times 9 \times 9 \times 9 \times 9 (b) \tfrac{2}{7} \times \tfrac{2}{7} \times \tfrac{2}{7} (c) 13^4
Solution(a) Every factor is 9, so 9 is the base. Count the factors: in pairs there are three pairs, so six factors, and the exponent is 6. The product is 9^6.
(b) Every factor is the fraction \tfrac{2}{7}, so the whole fraction is the base, and there are three factors. The power is \left(\tfrac{2}{7}\right)^3. The brackets are needed: \tfrac{2^3}{7} would mean \tfrac{2 \times 2 \times 2}{7}, with only the top cubed.
(c) The exponent 4 says four factors of the base 13: 13^4 = 13 \times 13 \times 13 \times 13. It is four thirteens multiplied, not 13 \times 4.
Two different factors
Write 3 \times 5 \times 3 \times 5 \times 3 using powers, then find its value.
SolutionThe factors are not all the same, so this is not a single power. The commutative property lets the factors be put in any order, so gather the threes and the fives: 3 \times 3 \times 3 \times 5 \times 5. The associative property lets them be grouped any way, so group each kind: (3 \times 3 \times 3) \times (5 \times 5). Each group is a power:
For the value, 3^3 = 27 and 5^2 = 25, and 27 \times 25 = 675.
Check. Multiply the original product from left to right: 3 \times 5 = 15, 15 \times 3 = 45, 45 \times 5 = 225, 225 \times 3 = 675. The same.
This is how a prime factorization is usually written: 675 = 3^3 \times 5^2. The exponents say at a glance how many of each prime there are, which is exactly what finding a GCF or LCM from prime factors reads off.
Round five of the telephone chain
In Tali’s chain, everyone who is called phones 3 new people. How many people are called in round five, and how many in the first five rounds altogether? Compare the first answer with 3 \times 5 and with 5^3.
SolutionEach round multiplies the count by 3, starting from 3 in round one, so round five has five factors of 3: 3^5. Work it out one factor at a time, keeping a running product. Multiplying from the left in pairs like this is allowed by the associative property.
Round five has 243 calls. The running products 3, 9, 27, 81 are the earlier rounds, so the first five rounds hold
calls in all.
Now compare. 3 \times 5 = 15 adds five threes, and 5^3 = 5 \times 5 \times 5 = 125 multiplies three fives. Neither is 243: the power is not the product of its two numbers, and swapping base and exponent changes the value.
Folding a sheet of paper
Wes folds a sheet of paper in half, then in half again, five times in all. How many layers thick is the folded paper? If each layer is 0.1 millimeters thick, how thick is the folded stack?
SolutionEach fold lays one half on top of the other, so it doubles the number of layers, as in Figure 2. Starting from one layer, five folds multiply by 2 five times:
The running products 2, 4, 8, 16, 32 are the layers after each fold, so there are 32 layers. Five folds do not add 5 layers or make 2 \times 5 = 10; each one doubles what is there.
The stack is 32 layers of 0.1 millimeters:
so it is 3.2 millimeters thick. Multiplying by 0.1 moves every digit one place to the right.
Powers of fractions
Find (a) \left(\tfrac{2}{3}\right)^2 and (b) \left(\tfrac{1}{2}\right)^5, and (c) compare \tfrac{2^2}{3} with the answer to (a).
Solution(a) Two factors of \tfrac{2}{3}. Multiply the tops and the bottoms:
Figure 3 shows it: \tfrac{2}{3} of \tfrac{2}{3} of the square is 4 of its 9 equal pieces. The answer is less than \tfrac{2}{3}, because taking two thirds of something leaves less of it.
(b) Five factors of \tfrac{1}{2}. The top is five ones multiplied, which is 1, and the bottom is 2^5, which Example 5 found to be 32:
This is Wes’s paper again. Each fold halves the area of the top face, so after five folds the top face is \tfrac{1}{32} of the sheet, and 32 layers of \tfrac{1}{32} of a sheet make one sheet.
(c) In \tfrac{2^2}{3} the exponent sits on the 2 only, so it is \tfrac{4}{3} = \tfrac{12}{9}, exactly three times \tfrac{4}{9}. The brackets decide what the base is.
Powers of decimals
Find 0.1^3, 0.3^2 and 1.2^2.
Solution0.1 is one tenth. A tenth of a tenth is a hundredth, so 0.1 \times 0.1 = 0.01, and a tenth of that is a thousandth:
Each factor of 0.1 moves the digits one place to the right, three factors three places. The answer has three decimal places, one from each factor.
For 0.3^2 = 0.3 \times 0.3, multiply the digits, 3 \times 3 = 9, and count places: tenths times tenths are hundredths, so two places, and 0.3^2 = 0.09. As fractions, \left(\tfrac{3}{10}\right)^2 = \tfrac{9}{100}, which agrees. The value 0.9 would be nine tenths, ten times too large.
For 1.2^2 = 1.2 \times 1.2, the digits give 12 \times 12 = 144, and one place from each factor makes two, so 1.2^2 = 1.44.
Notice the sizes: 1.44 is more than 1.2, because the base is more than 1, while 0.09 is less than 0.3, because the base is between 0 and 1.
Boundaries
The exponent counts, so here it is a counting number. An exponent of 1, 2, 3 and so on says how many factors there are. An exponent of 0 would count no factors at all, so 5^0 needs a meaning of its own, which Zero and negative exponents sets. A fraction as an exponent cannot count factors either and needs a new meaning, which comes much later.
The exponent reaches only the base written just before it. \left(\tfrac{2}{3}\right)^2 and \tfrac{2^2}{3} are different numbers, and in 2 \times 5^2 only the 5 is squared, so it is 2 \times 25 = 50. How powers sit inside longer expressions is Order of operations with powers.
A power has one base. Only a product whose factors are all the same is a single power. A product of different numbers, such as 3^3 \times 5^2, keeps one power for each base.
Base and exponent play different parts. Swapping them usually changes the value, as 3^5 = 243 and 5^3 = 125 showed.
The bases here are never negative. Every base in this lesson is a whole number, a fraction or a decimal, none below zero. A negative base raises questions about signs and brackets that Order of operations with negative numbers takes up.
Where this goes next
The numbers that are second and third powers of whole numbers get their own names and pictures in Perfect squares and perfect cubes, and powers start appearing inside longer calculations in Order of operations with powers, where they are worked out before multiplying.
Powers of ten are the place-value columns, which is the subject of Powers of ten. Writing a very large or very small number as a decimal times a power of ten is Scientific notation.
The zero exponent asks what 5^0 should be, and answers it by keeping the patterns of this lesson going. Later, counting factors gives the first rule for working with powers of the same base: 2^3 \times 2^4 is three twos and four more, seven in all, which is multiplying powers.
And the telephone chain is the first glimpse of growth by a constant factor, which becomes exponential functions.
Exercises
Exercises 1 to 9 go with the first part of the lesson (Examples 1 to 3), Exercises 10 to 22 with the second (Examples 4 to 7), and the reasoning group draws on both.
Part 1: naming and reading powers
For Exercises 1 to 3, name the base and the exponent, write the power in expanded form, and say it aloud.
- 7^4
- 12^2
- \left(\tfrac{5}{6}\right)^3
- Write in exponential form: “eight cubed” and “three to the seventh power”.
Part 1: repeated products and powers
Write Exercises 5 to 7 as powers.
- 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4
- 0.6 \times 0.6 \times 0.6 \times 0.6
- \tfrac{5}{8} \times \tfrac{5}{8}
- Write 15^3 in expanded form.
- Write 7 \times 2 \times 7 \times 7 \times 2 using powers, then find its value.
Part 2: the value of a power
Find the value.
- 5^4
- 9^3
- 1^8
- 0^6
- 6^1
- Which is larger, 2^8 or 8^2?
Part 2: fractions and decimals
Find the value.
- \left(\tfrac{3}{4}\right)^2
- \left(\tfrac{2}{5}\right)^3
- 0.2^3
- 1.1^2
- 0.05^2
Part 2: chains and folds
- Yuri starts a telephone chain in which everyone who is called phones 4 new people. How many people are called in round 4? How many are called in the first four rounds altogether?
- Zev folds a sheet of paper in half again and again. Suppose the paper could be folded as many times as he liked. What is the fewest number of folds that gives more than 1000 layers?
Reasoning
- 2^4 and 4^2 are both 16. Does swapping the base and the exponent always leave the value the same? Explain, and say why 2^4 and 4^2 happen to agree.
- Explain why 10^6 is written as a 1 followed by six zeros.
- Show that \left(\tfrac{7}{8}\right)^2 is less than \tfrac{7}{8} but \left(\tfrac{9}{8}\right)^2 is more than \tfrac{9}{8}, and explain why.
Answers
- Base 7, exponent 4: 7 \times 7 \times 7 \times 7. Say “seven to the fourth power”, or “seven to the fourth”.
- Base 12, exponent 2: 12 \times 12. Say “twelve squared”.
- Base \tfrac{5}{6}, exponent 3: \tfrac{5}{6} \times \tfrac{5}{6} \times \tfrac{5}{6}. Say “five sixths cubed”. The brackets make the whole fraction the base.
- 8^3 and 3^7. Cubed means an exponent of 3, and the seventh power an exponent of 7.
- 4^7. There are seven factors, each 4.
- 0.6^4. Four factors, each 0.6.
- \left(\tfrac{5}{8}\right)^2. The whole fraction is the base, so it goes in brackets.
- 15 \times 15 \times 15. Three factors of 15.
- 2^2 \times 7^3 = 1372. Reorder (commutative property) and group (associative property): (2 \times 2) \times (7 \times 7 \times 7). Then 2^2 = 4, 7^3 = 343 and 4 \times 343 = 1372.
- 625. The running products are 5, 25, 125, 625.
- 729. The running products are 9, 81, 729.
- 1. Every factor is 1, so the product is 1.
- 0. A product with a factor of 0 is 0.
- 6. An exponent of 1 means one factor, the base itself.
- 2^8. 2^8 = 256, the running products being 2, 4, 8, 16, 32, 64, 128, 256, while 8^2 = 64.
- \tfrac{9}{16}. Tops 3 \times 3 = 9, bottoms 4 \times 4 = 16.
- \tfrac{8}{125}. Tops 2 \times 2 \times 2 = 8, bottoms 5 \times 5 \times 5 = 125.
- 0.008. The digits give 2 \times 2 \times 2 = 8, and three factors with one decimal place each give three places.
- 1.21. The digits give 11 \times 11 = 121, and two factors with one place each give two places.
- 0.0025. The digits give 5 \times 5 = 25, and two factors with two places each give four places. As fractions, \left(\tfrac{5}{100}\right)^2 = \tfrac{25}{10000}.
- 256 in round 4, and 340 in the first four rounds. Each round multiplies by 4, so round four has 4^4 = 4 \times 4 \times 4 \times 4 = 256. The rounds hold 4, 16, 64 and 256, and 4 + 16 + 64 + 256 = 340.
- 10 folds. Each fold doubles the layers, so the number of folds is the exponent on 2. The layers after each fold are 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. So 2^9 = 512 is still less than 1000 and 2^{10} = 1024 is more.
- No. For example 2^3 = 8 but 3^2 = 9. The base says what is multiplied and the exponent how many times, so swapping them changes both. 2^4 and 4^2 agree because four twos can be grouped in pairs: 2 \times 2 \times 2 \times 2 = (2 \times 2) \times (2 \times 2) = 4 \times 4.
- 10^6 is six factors of ten. Starting from 1, each multiplication by 10 moves every digit one place to the left and leaves a 0 in the ones place. Six factors do this six times, giving 1 followed by six zeros, 1{,}000{,}000.
- \left(\tfrac{7}{8}\right)^2 = \tfrac{49}{64}, which is less than \tfrac{7}{8} = \tfrac{56}{64}. \left(\tfrac{9}{8}\right)^2 = \tfrac{81}{64}, which is more than \tfrac{9}{8} = \tfrac{72}{64}. Squaring \tfrac{7}{8} takes \tfrac{7}{8} of \tfrac{7}{8}, only part of it, so less is left. Squaring \tfrac{9}{8} takes all of \tfrac{9}{8} and an eighth of it more.