Divide by Powers of Ten and Explain the Pattern | Grade 5 Math | LevelKite

You already know that multiplying a decimal by a power of ten moves each digit to the left. Dividing undoes multiplying. So you might guess that dividing moves each digit back to the right. This lesson shows why that guess is correct.

First, a quick reminder of three math terms. A power of ten is a number you can make by multiplying only 10s: $10$ , then $10 \times 10 = 100$ , then $10 \times 10 \times 10 = 1{,}000$ . Each 10 that is multiplied is a factor of 10. The exponent is the small raised number that counts those factors: $10^1 = 10$ , $10^2 = 100$ and $10^3 = 1{,}000$ .

Now think about what dividing by 10 means. It makes a number one tenth as big. That happens to every part of the number. So 5 tens become 5 ones, 2 ones become 2 tenths, and 7 tenths become 7 hundredths. The table shows this for $52.7 \div 10$ .

Each digit of 52.7 before and after dividing by 10

Digit	Place in 52.7	Value in 52.7	Place after dividing	Value after dividing
5	tens	$50$	ones	$5$
2	ones	$2$	tenths	$0.2$
7	tenths	$0.7$	hundredths	$0.07$

Read across each row. Every digit shifts, or moves, one place to the right, and its value becomes one tenth of what it was. Add the new values: $5 + 0.2 + 0.07 = 5.27$ . So $52.7 \div 10 = 5.27$ . The digits are still 5, 2 and 7, in the same order. Only their places changed.

What if you divide by $10^2$ ? Because $10^2 = 10 \times 10$ , dividing by $10^2$ is the same as dividing by 10 two times: first $52.7 \div 10 = 5.27$ , and then $5.27 \div 10 = 0.527$ . Each division shifts every digit one more place to the right. Two factors of 10 make two places, so $52.7 \div 10^2 = 0.527$ . The ones place has no digit now, so you write a 0 there.

Key concept

Digits shift right; the decimal point stays

Dividing by 10 makes each digit worth one tenth as much, so every digit shifts one place to the right. You do that once for each factor of 10, and the exponent counts the factors. Dividing by $10^1$ moves each digit 1 place. Dividing by $10^2$ moves each digit 2 places. Dividing by $10^3$ moves each digit 3 places.

The decimal point does not move. It always sits between the ones place and the tenths place.

Sometimes the shift leaves the ones place, or places right after the decimal point, with no digit. Write a 0 in each of those places. Each 0 is a placeholder: a zero that holds a place so the other digits stay where they belong.

Worked example

Shift two places to the right

What is $128.4 \div 10^2$ ?

Count the factors of 10. The exponent is 2, so there are two factors of 10. Each factor moves every digit one place to the right, so every digit moves two places.
$10^2 = 10 \times 10 = 100$
Move each digit two places to the right. The 1 goes from hundreds to ones. The 2 goes from tens to tenths. The 8 goes from ones to hundredths. The 4 goes from tenths to thousandths. Each value is now one hundredth of what it was.
$100 + 20 + 8 + 0.4 \;\to\; 1 + 0.2 + 0.08 + 0.004$
Write the digits in their new places. Keep the decimal point between the ones place and the tenths place.
$128.4 \div 10^2 = 1.284$
Check that the size makes sense. 128.4 is a little more than 100, and 100 divided by 100 is 1. So the answer should be a little more than 1, and 1.284 is.

Answer

$128.4$ divided by $10^2$ is $1.284$ .

Worked example

Write zeros as placeholders

What is $3.5 \div 10^2$ ?

Count the factors of 10. The exponent is 2, so every digit moves two places to the right.
$10^2 = 10 \times 10$
Move each digit. The 3 goes from ones to hundredths. The 5 goes from tenths to thousandths.
$3 + 0.5 \;\to\; 0.03 + 0.005$
Look for empty places. No digit is in the ones place or the tenths place now. Write a 0 in each one as a placeholder. Those zeros keep the 3 in the hundredths place and the 5 in the thousandths place.
$3.5 \div 10^2 = 0.035$
Check by multiplying back. Multiplying by the same power of ten moves each digit two places to the left, back to where it started.
$0.035 \times 10^2 = 3.5$

Answer

$3.5$ divided by $10^2$ is $0.035$ . The zeros in the ones place and the tenths place are placeholders.

Now look at where the decimal point ends up. Compare $3.5$ with $0.035$ . It can look as if the decimal point jumped two spots to the left. It did not move at all.

The decimal point has one job. It marks the line between the ones place and the tenths place, and that line is in the same spot in every number. Even a whole number such as $907$ has that line, right after the 7. You just do not write the point.

The chart below puts $3.5$ , then $3.5 \div 10$ , then $3.5 \div 10^2$ into place-value columns. Look down the decimal point column first. Then follow the 3 and the 5 from row to row.

A dash means no digit is written in that place.

Number	Ones	Decimal point	Tenths	Hundredths	Thousandths
$3.5$	3	.	5	—	—
$3.5 \div 10 = 0.35$	0	.	3	5	—
$3.5 \div 10^2 = 0.035$	0	.	0	3	5

The decimal point is in the same column in every row. It stayed still. The 3 and the 5 did the moving: one column to the right for each factor of 10. In the bottom row, zeros hold the ones place and the tenths place, where no digit is left.

So the pattern has a simple reason. The digits change places, and the decimal point stays between the ones and the tenths. To explain any answer, name a digit, its old place and its new place. For example: "The 3 moved from the ones place to the hundredths place, so $3.5 \div 10^2 = 0.035$ ."

You can use the same ideas to check work with powers of ten, for multiplying as well as dividing. Three kinds of mistakes come up often, so ask three questions:

Which way? Dividing by a power of ten makes a number smaller, so the digits move right. Multiplying makes it bigger, so the digits move left.
How many places? The exponent counts the factors of 10. That is how many places each digit moves.
Did the digits really move? Pick one digit and name its place before and after. Writing extra zeros is not the same as moving a digit.

Worked example

Find and fix a mistake

A student wrote $45.2 \div 10^2 = 4{,}520$ . Is this correct? If not, find the mistake and fix it.

Ask which way. Dividing by a power of ten makes a number smaller, so the answer must be less than 45.2. The student's answer is much bigger than 45.2, so it cannot be correct.
$4{,}520 > 45.2$
Follow one digit to find the mistake. The 4 started in the tens place. In the student's answer it is in the thousands place, two places to the left. Moving left is what multiplying does, so the student multiplied instead of dividing.
$45.2 \times 10^2 = 4{,}520$
Fix it. The exponent is 2, so move each digit two places to the right. 4 tens become 4 tenths, 5 ones become 5 hundredths, and 2 tenths become 2 thousandths. The ones place is empty, so write a 0 there.
$45.2 \div 10^2 = 0.452$
Check by multiplying back. Each digit moves two places to the left and lands where it started.
$0.452 \times 10^2 = 45.2$

Answer

The work is not correct. The digits moved the wrong way. The correct answer is $45.2 \div 10^2 = 0.452$ .

Three more mistakes, checked with the three questions

Work with a mistake	What went wrong	Correct work
$0.9 \times 10^2 = 0.009$	Wrong way. Multiplying moves digits to the left, but the 9 moved to the right.	$0.9 \times 10^2 = 90$
$907 \div 10^3 = 9.07$	Wrong number of places. The exponent is 3, but each digit moved only 2 places.	$907 \div 10^3 = 0.907$
$0.26 \times 10^2 = 0.2600$	Zeros counted, not places. Two zeros were added, but no digit moved. The value is still $0.26$ .	$0.26 \times 10^2 = 26$

Here is the idea to keep. Dividing by a power of ten makes every digit worth less, so each digit moves one place to the right for each factor of 10. The decimal point never moves. It stays between the ones and the tenths. If the shift empties the ones place or the places just after the decimal point, zeros hold those places. To check any answer, ask three things: Which way? How many places? Did the digits really move?