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Exponents and expressions

Write repeated multiplication as a power, evaluate powers, and use them in the order of operations.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

In this lesson you write repeated multiplication with an exponent, read and evaluate powers of whole numbers, fractions and decimals, use powers of 10 to understand place value, and work out expressions that contain powers in the right order: parentheses, exponents, multiplication and division, then addition and subtraction.

2. What you already know

You can multiply whole numbers, decimals and fractions. You know that multiplication is repeated addition: $4 \times 3$ is $3 + 3 + 3 + 3$. You found the area of a square by multiplying its side by itself and the volume of a cube by multiplying its edge three times. In grade 5 you saw powers of 10, such as $10^3 = 1{,}000$, when you studied place value. You also know the order of operations: parentheses first, multiplying and dividing before adding and subtracting. This lesson gives a short way to write repeated multiplication and shows where it fits in that order.

3. Words this lesson uses

4. A short way to write repeated multiplication

Multiplication was invented as a shortcut for adding the same number again and again. Exponents are the next shortcut: they write multiplying the same number again and again. Instead of $2 \times 2 \times 2 \times 2 \times 2$, we write

$$2^5,$$

read 'two to the fifth power' or 'two to the fifth'. The 2 is the base, the number being multiplied. The 5 is the exponent, which counts how many factors of the base there are. The value is $2^5 = 32$.

The most important thing to understand is that the exponent is a count, not a factor. $2^5$ is not $2 \times 5 = 10$. It is five 2s multiplied together, which is 32. The difference grows fast: $10 \times 6 = 60$, but $10^6$ is one million.

Two powers have special names that come from geometry. A square with sides of 7 units has an area of $7 \times 7 = 7^2 = 49$ square units, so $7^2$ is read 'seven squared'. A cube with edges of 4 units has a volume of $4 \times 4 \times 4 = 4^3 = 64$ cubic units, so $4^3$ is read 'four cubed'.

5. Reading, writing and evaluating powers

To write a product as a power, count the factors. $6 \times 6 \times 6 \times 6$ has four factors of 6, so it is $6^4$. A product with two different numbers uses two powers: $2 \times 2 \times 2 \times 7 \times 7 = 2^3 \times 7^2$.

To evaluate a power, write it out and multiply one factor at a time. Keep a count so you use exactly the right number of factors:

$$3^5 = 3 \times 3 \times 3 \times 3 \times 3: \quad 9, \; 27, \; 81, \; 243.$$

Each number in the list is the one before it times 3. Two more facts help:

• A power with exponent 1 is just the base: $8^1 = 8$, one factor of 8.
• The base 1 never grows: $1^{50} = 1$, because 1 times 1 is always 1.

Learning the first few squares and cubes by heart saves time: the squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and the cubes 1, 8, 27, 64, 125.

Say powers out loud carefully, because the words carry the meaning. $5^2$ is 'five squared' or 'five to the second power'. $5^3$ is 'five cubed' or 'five to the third power'. From 4 on, use the ordinal: $5^4$ is 'five to the fourth power', $5^7$ is 'five to the seventh'. Writing a power from words works the other way round: 'nine to the sixth power' is $9^6$, the 9 on the line and the 6 raised. A calculator key marked with a caret, ^, does the same job: typing 5 ^ 3 gives 125. The caret is also how people write powers in an email or a spreadsheet, where raised numbers are hard to type.

6. Powers of 10 and place value

Powers of 10 are the easiest powers to evaluate and the most useful. Each factor of 10 adds one zero:

So the exponent of a power of 10 is the number of zeros. This is why our place value system works: the places are ones, tens ($10^1$), hundreds ($10^2$), thousands ($10^3$) and so on. A number like 4,000 is $4 \times 10^3$, and a billion, 1,000,000,000, is $10^9$. Scientists use powers of 10 to write very large numbers without long strings of zeros.

7. Fractions and decimals as bases

The base does not have to be a whole number. The exponent still counts factors.

$$\left(\frac{2}{3}\right)^2 = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9}, \qquad 0.5^3 = 0.5 \times 0.5 \times 0.5 = 0.125.$$

Notice two things. First, parentheses matter with a fraction: $\left(\tfrac{2}{3}\right)^2$ squares the whole fraction, top and bottom. Second, a base between 0 and 1 gets smaller as the exponent grows, because multiplying by a number less than 1 shrinks: $0.5^1 = 0.5$, $0.5^2 = 0.25$, $0.5^3 = 0.125$. That is the opposite of whole-number bases, which grow. For decimals, count decimal places: $0.3^2 = 0.09$ has $1 + 1 = 2$ decimal places, because it is $0.3 \times 0.3$.

8. Powers in the order of operations

In an expression, exponents are worked out right after parentheses and before any multiplying, dividing, adding or subtracting:

1. Parentheses
2. Exponents
3. Multiplication and division, from left to right
4. Addition and subtraction, from left to right

An exponent applies only to the number or the parentheses directly in front of it. In $5 \times 2^3$ the exponent belongs to the 2, so the value is $5 \times 8 = 40$, not $10^3 = 1{,}000$. In $(5 \times 2)^3$ the parentheses make the whole product the base, so the value is $10^3 = 1{,}000$.

A sum in parentheses works the same way: $(3 + 4)^2 = 7^2 = 49$, while $3^2 + 4^2 = 9 + 16 = 25$. Squaring a sum is not the same as adding the squares. Draw a square 7 units on a side, split each side into 3 and 4, and you see the two small squares plus two rectangles of $3 \times 4$: $9 + 16 + 12 + 12 = 49$.

9. The method, step by step, and how to check it

To work with exponents:

1. Find the base and the exponent. Check whether parentheses make a whole expression the base.
2. Write the power as repeated multiplication if you are not sure of its value.
3. Multiply one factor at a time, counting as you go, until the count equals the exponent.
4. In a longer expression, follow the order: parentheses, exponents, multiplication and division, addition and subtraction.
5. Write the answer and, in a story, its unit: square units for a squared length, cubic units for a cubed one.

To check:

• Count the factors in your written-out product. There must be exactly as many as the exponent.
• A whole-number base bigger than 1 must give a value bigger than base times exponent once the exponent is 3 or more: $2^3 = 8 > 6$.
• The last digit follows a pattern: every power of 5 ends in 5, every power of 10 ends in 0, and powers of 2 end in 2, 4, 8, 6, over and over.
• For a power of 10, count the zeros: they must equal the exponent.

10. In the world: bits and bytes in a computer

A computer stores everything as bits, and each bit is either 0 or 1: two choices. Two bits together have $2 \times 2 = 2^2 = 4$ possible patterns: 00, 01, 10 and 11. Every extra bit doubles the number of patterns, so a byte, which is 8 bits, has

$$2^8 = 256$$

patterns. That is why a single byte can store any whole number from 0 to 255, and why many older video games allowed at most 255 of something. Memory sizes use powers of 2 as well: $2^{10} = 1{,}024$, which is close to one thousand, so 1,024 bytes has long been called a kilobyte. A 16-bit number has $2^{16} = 65{,}536$ patterns, and each time engineers added bits, the numbers a computer could handle grew by a power of 2, not by a little.

11. In the world: ordering concrete by the cubic yard

In the United States, ready-mix concrete is sold by the cubic yard. A yard is 3 feet, so a cubic yard is a cube 3 feet on every edge, and its volume in cubic feet is

$$3^3 = 3 \times 3 \times 3 = 27 \text{ cubic feet}.$$

A patio slab 12 feet by 9 feet and $\tfrac{1}{3}$ foot (4 inches) thick has a volume of $12 \times 9 \times \tfrac{1}{3} = 36$ cubic feet. Dividing by 27 gives $1\tfrac{1}{3}$ cubic yards, so the builder orders about $1\tfrac{1}{2}$ cubic yards to allow a little extra. A builder who thought a cubic yard was only $3 \times 3 = 9$ cubic feet would order three times too little concrete. The exponent 3 is there because a cube has three edges that each grew by a factor of 3.

12. Mistakes to watch for

Multiplying the base by the exponent. $4^3$ is $4 \times 4 \times 4 = 64$, not $4 \times 3 = 12$.

Adding instead of multiplying. $4^3$ is not $4 + 4 + 4 = 12$ either; that is $3 \times 4$.

Giving the exponent to the wrong number. In $2 \times 5^2$ only the 5 is squared: $2 \times 25 = 50$, not $10^2 = 100$.

Squaring each part of a sum. $(2 + 3)^2 = 25$, not $4 + 9 = 13$.

Doing the power last. In $6 + 2^3$, find $2^3 = 8$ before adding: the value is 14, not $8^3$.

Counting zeros wrong. $10^4$ has four zeros, 10,000, not 1,000.

13. Evaluating a power one factor at a time

1. Evaluate $3^4$. Name the base and the exponent.

   $\text{base} = 3, \qquad \text{exponent} = 4$

   The base is multiplied, and the exponent counts how many times it appears as a factor.

2. Write the power as repeated multiplication.

   $3^4 = 3 \times 3 \times 3 \times 3$

   Four factors of 3, one for each count of the exponent.

3. Multiply the first two factors.

   $3 \times 3 = 9$

   Working from the left keeps track of how many factors are used: two so far.

4. Multiply by the third factor.

   $9 \times 3 = 27$

   Three factors used; this is $3^3$.

5. Multiply by the fourth factor.

   $27 \times 3 = 81$

   Four factors used, so $3^4 = 81$, far more than $3 \times 4 = 12$.

14. Writing a product with two bases

1. Write $2 \times 2 \times 2 \times 5 \times 5$ with exponents. Count the factors of 2.

   $2 \times 2 \times 2 = 2^3$

   Three factors of 2 make the power $2^3$.

2. Count the factors of 5.

   $5 \times 5 = 5^2$

   Two factors of 5 make $5^2$.

3. Write the whole product with exponents.

   $2^3 \times 5^2$

   Different bases get their own powers, joined by multiplication.

4. Evaluate the first power.

   $2^3 = 8$

   Two times two is four, and four times two is eight.

5. Evaluate the second power.

   $5^2 = 25$

   Five squared is the area of a 5-by-5 square.

6. Multiply the two values.

   $8 \times 25 = 200$

   Four 25s make 100, so eight 25s make 200; multiplying the five original factors gives the same.

15. Powers inside a longer expression

1. Evaluate $4 + 3 \times (5 - 2)^2 - 2^3$. Work inside the parentheses.

   $5 - 2 = 3$

   Parentheses come first.

2. Square the result of the parentheses.

   $3^2 = 9$

   The exponent 2 belongs to the whole parentheses.

3. Evaluate the other power.

   $2^3 = 8$

   All exponents are done before any multiplying or adding.

4. Rewrite the expression with the powers replaced.

   $4 + 3 \times 9 - 8$

   Only multiplication, addition and subtraction are left.

5. Multiply the 3 by the square.

   $3 \times 9 = 27$

   Multiplication comes before addition and subtraction.

6. Add and subtract from left to right.

   $4 + 27 - 8 = 31 - 8 = 23$

   Addition and subtraction are done in the order they appear.

7. Check against a common wrong order.

   $(4 + 3) \times 9 - 8 = 55 \ne 23$

   Adding the 4 too early gives a different value, so the order of operations matters.

16. Your turn: evaluate $\left(\tfrac{1}{2}\right)^4$

1. Write the power as repeated multiplication.

   $\tfrac{1}{2} \times \tfrac{1}{2} \times \tfrac{1}{2} \times \tfrac{1}{2}$

   The exponent 4 counts four factors of one half.

2. Multiply the tops and the bottoms.

   $\dfrac{1 \times 1 \times 1 \times 1}{2 \times 2 \times 2 \times 2}$

   Fractions multiply top with top and bottom with bottom.

3. Your turn: work this step out. Its working is at the end of the packet.

   Evaluate the top and the bottom.

17. Guided practice

A student says $9^4$ means $9 \times 4 = 36$. Which explanation corrects the mistake?

• The exponent counts how many factors of the base are multiplied together.
• The exponent is multiplied by the base once.
• The base is added to itself as many times as the exponent says.

18. Guided practice

Evaluate $(3 + 8)^2 - 3^2$. Complete the worked solution.

1. Work inside the parentheses.

   $3 + 8 =$ _s

   Parentheses come first in the order of operations.

2. Square the sum.

   $\text{sum} \times \text{sum} =$ _p

   The exponent 2 means two factors of the whole number in the parentheses.

3. Square the number being subtracted.

   $3 \times 3 =$ _q

   Exponents come before subtraction, so this square is found before subtracting.

4. Subtract the second square from the first.

   $\text{first square} - \text{second square} =$ _r

   Subtraction comes last, once every power is a plain number.

5. Check that squaring each part and adding would not do.

   $(3 + 8)^2 \ne 3^2 + 8^2$

   Squaring a sum makes a square with extra rectangles in it, so it is bigger than the two small squares.

19. Guided practice

Write 4 × 4 × 4 using an exponent, and give its value.

4 × 4 × 4 is 4 to the power _e, and its value is _v.

20. Practice

Evaluate $3^3$. Give the value of the first two factors multiplied together, then the value of the whole power.

The first two factors make _sq, and the whole power is _answer.

21. Practice

Evaluate $3 + 5^2 \times 4$. Give the power, the product and the final value.

The power is _pw, the product is _pr, and the value is _answer.

22. Practice

Complete the table with the exponent and the value for each power. Row 1: the area, in square inches, of a square tile 11 inches on a side. Row 2: the volume, in cubic feet, of a cube-shaped crate 6 feet on an edge. Row 3: the number 1 followed by 6 zeros, written as a power of 10.

23. Somewhere new

A bike lock has 5 dials in a row, and each dial can be turned to any one of 7 different colors. How many settings would a lock with just the first two dials have, and how many settings does the whole lock have?

Two dials give _two settings, and the whole lock has _answer settings.

24. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

25. Test question

In a lab, a culture of bacteria doubles every 20 minutes. It starts with 12 cells. How many doublings happen in 60 minutes, what is the growth factor as a number, and how many cells are there at the end?

There are _d doublings, the growth factor is _f, and the culture ends with _c cells.

26. What you can do now

You can write and evaluate powers and use them in expressions. Without looking: why is $2^5$ equal to 32 and not 10, and what is $3 + 2 \times 4^2$?

Working for the steps left to you

16. Your turn: evaluate $\left(\tfrac{1}{2}\right)^4$, step 3