Virginia SOL 6 - 2020 Edition 1.11 Place value and powers of 10
Lesson

## Powers of ten

Recall that an exponent (or power) tells us the number of times to multiply a certain number by itself. Let's review the patterns in the powers of ten with the following exploration.

#### Exploration

Move the slider in the applet below and think about the answers to the following questions.

1. What patterns do you see between the power of ten and its expanded form?
2. What patterns do you see between the power of ten and the number of zeros?
3. Why do you think $10^0=1$100=1?

Notice that the power of ten is the same as the number of zeros before or after the decimal point.  That's because of the place value system.

The following table demonstrates another way to think of some of the powers of ten that we see in the applet.

Power of Ten Meaning Value (basic numeral) In Words
$10^5$105 $10\times10\times10\times10\times10$10×10×10×10×10 $100000$100000 One hundred thousand
$10^4$104 $10\times10\times10\times10$10×10×10×10 $10000$10000 Ten thousand
$10^3$103 $10\times10\times10$10×10×10 $1000$1000 One thousand
$10^2$102 $10\times10$10×10 $100$100 One hundred
$10^1$101 $10$10 $10$10 Ten
$10^0$100 $1$1 $1$1 One

#### Worked example

##### Question 1

What value should go in the space?

$300=\editable{}\times10^2$300=×102

Think: What basic numeral is equivalent to $10^2$102? How will this move our decimal place?

Do:

$10^2$102 is equivalent to $10\times10$10×10 or $100$100. So we can rewrite the question as:

$300=\editable{}\times100$300=×100

So the missing value is $3$3 because $3\times100=300$3×100=300.

Reflect:

This means that $300$300 is $3\times10^2$3×102.

#### Practice questions

##### Question 4

Complete the following expression.

1. $4000$4000$=$= $\editable{}$ × $10^3$103

##### Question 5

The distance between two stars is approximate $9\times10^7$9×107 meters.

Express this distance as a whole number.

## Rewriting numbers using powers of ten

When we need to write our number using expanded form and powers of ten, we can tackle this in two steps.

### Step 1 - Write in expanded form

To write a number in expanded form means we are breaking our number into separate 'chunks', where each chunk represents the value of each digit.  So, think of a number like $24$24. This has $2$2 tens, and $4$4 units, so would be $20+4$20+4, in expanded form.  We make sure that we distribute every digit, so thinking of place value is really helpful here.

Number Expanded form
$3468$3468 $3000+400+60+8$3000+400+60+8
$23179$23179 $20000+3000+100+70+9$20000+3000+100+70+9

### Step 2 - Express as powers of ten

Our next step involves looking at each chunk of our expanded form and considering if we can express it as a power of ten.

#### Practice questions

##### Question 6

Express $2487$2487 in the expanded form using exponential notation. For example, $245$245 in this form is equal to $2\times10^2+4\times10^1+5\times1$2×102+4×101+5×1.

##### Question 7

Write the number expressed by: $2\times10^5+5\times10^4+6\times10^3+7\times10^1$2×105+5×104+6×103+7×101.

Note: Do not use comma separators

### Outcomes

#### 6.4

Recognize and represent patterns with whole number exponents and perfect squares