Multiply Whole Numbers and Decimals by Powers of Ten | Grade 5 Math | LevelKite

What does multiplying by 10 really do? It makes every part of a number ten times as big. So 4 ones become 4 tens, and 3 tens become 3 hundreds.

In our number system, each place is worth ten times the place on its right. So multiplying by 10 moves, or shifts, every digit one place to the left. A digit is a single symbol in a number, like the 3 and the 4 in $34$ .

You have already met powers of ten: numbers such as $10$ , $100$ and $1{,}000$ , built by multiplying 10s. The numbers you multiply are called factors. So $10$ has one factor of 10, $100 = 10 \times 10$ has two, and $1{,}000 = 10 \times 10 \times 10$ has three.

Each factor of 10 shifts every digit one more place to the left. The chart below shows this for the whole number $34$ .

Multiplying 34 by powers of ten. A dash (–) means no digit is in that place.

Number sentence	Ten thousands	Thousands	Hundreds	Tens	Ones
$34$ (start)	–	–	–	3	4
$34 \times 10 = 340$	–	–	3	4	0
$34 \times 100 = 3{,}400$	–	3	4	0	0
$34 \times 1{,}000 = 34{,}000$	3	4	0	0	0

Read the chart one row at a time. In $34$ , the 3 is in the tens place and the 4 is in the ones place. So $34$ is 3 tens and 4 ones.

Multiply by $10$ , and each digit shifts one place to the left. Now the 3 is in the hundreds place and the 4 is in the tens place. That makes 3 hundreds and 4 tens, so the product (the answer to a multiplication) is $340$ .

Multiply by $100$ , and each digit shifts two places, because $100$ has two factors of 10. Now the 3 is in the thousands place and the 4 is in the hundreds place, so the product is $3{,}400$ .

Multiply by $1{,}000$ , and each digit shifts three places. The product is $34{,}000$ .

Worked example

A whole number times 100

What is $275 \times 100$ ?

Count the factors of 10 in 100. There are two. Each factor of 10 shifts every digit one place to the left, so every digit shifts two places.
$100 = 10 \times 10$
See where each digit starts. In 275, the 2 is in the hundreds place, the 7 is in the tens place and the 5 is in the ones place.
$275 = 200 + 70 + 5$
Shift each digit two places to the left. The 2 moves to the ten thousands place, the 7 moves to the thousands place and the 5 moves to the hundreds place. Each digit is now worth 100 times as much.
$20{,}000 + 7{,}000 + 500$
The tens place and the ones place are now empty. Write a 0 in each one as a placeholder. The zeros keep the 2, the 7 and the 5 in their new places.
$275 \times 100 = 27{,}500$

Answer

The product of $275$ and $100$ is $27{,}500$ .

A decimal is a number written with a decimal point, such as $2.57$ . You multiply a decimal by a power of ten in exactly the same way. Every digit shifts one place to the left for each factor of 10. That includes the digits to the right of the decimal point.

Look at $2.57 \times 10$ . The 2 ones become 2 tens. The 5 tenths become 5 ones. The 7 hundredths become 7 tenths. So $2.57 \times 10 = 25.7$ .

Multiply by $100$ instead, and each digit shifts two places: $2.57 \times 100 = 257$ .

Now notice what stays still in the chart below. The decimal point always sits between the ones place and the tenths place. It can look as if the point jumped to the right. Really, the digits shifted to the left past it.

The decimal point sits between the Ones column and the Tenths column in every row.

Number sentence	Hundreds	Tens	Ones	Tenths	Hundredths
$2.57$ (start)	–	–	2	5	7
$2.57 \times 10 = 25.7$	–	2	5	7	–
$2.57 \times 100 = 257$	2	5	7	–	–

Worked example

A decimal times 100

What is $4.6 \times 100$ ?

Count the factors of 10. The number 100 has two, so every digit shifts two places to the left.
$100 = 10 \times 10$
Shift each digit. The 4 moves from the ones place to the hundreds place, so it is now worth 400. The 6 moves from the tenths place to the tens place, so it is now worth 60.
$400 + 60$
The ones place is now empty, so write a 0 there as a placeholder. This is not the same as writing zeros on the end of 4.6, which would shift no digit. This zero fills the gap that the shift left. It keeps the 4 in the hundreds place and the 6 in the tens place.
$4.6 \times 100 = 460$

Answer

The product of $4.6$ and $100$ is $460$ .

Key concept

Each factor of 10 shifts every digit one place left

To multiply a whole number or a decimal by a power of ten, shift every digit one place to the left for each factor of 10. Multiplying by $10$ shifts each digit one place. Multiplying by $100$ shifts each digit two places. Multiplying by $1{,}000$ shifts each digit three places.

The decimal point stays between the ones place and the tenths place.

Zeros only hold places. After the shift, write a $0$ in each empty place from the ones place up to the shifted digits.

You have also written powers of ten with an exponent. The exponent is the small raised number, like the 3 in $10^3$ . It counts the factors of 10. So $10^3$ means $10 \times 10 \times 10$ , which is $1{,}000$ .

Now use that in a product. $36 \times 1{,}000$ and $36 \times 10^3$ are two ways to write the same product.

The exponent also tells you how far the digits shift. An exponent of 3 means three factors of 10, so every digit shifts three places to the left.

To evaluate a product means to find its value. To evaluate $36 \times 10^3$ , shift the 3 and the 6 three places to the left. Then write a placeholder zero in each of the three empty places.

36 \times 1{,}000 = 36 \times 10^3 = 36{,}000

Worked example

Write it with an exponent, then evaluate

Write $3.08 \times 1{,}000$ using an exponent. Then evaluate it.

Write 1,000 as a power of ten with an exponent. 1,000 is three 10s multiplied together, so it has three factors of 10 and the exponent is 3.
$1{,}000 = 10 \times 10 \times 10 = 10^3$
Put this power of ten into the product in place of 1,000. The value stays the same, because both ways of writing name the same number.
$3.08 \times 1{,}000 = 3.08 \times 10^3$
Now evaluate. The exponent 3 means every digit shifts three places to the left. The 3 moves from the ones place to the thousands place. The 0 moves from the tenths place to the hundreds place. The 8 moves from the hundredths place to the tens place.
$3{,}000 + 0 + 80$
The ones place is now empty, so write a 0 there as a placeholder. The 0 that was already in 3.08 is a digit too. It shifted with the others and now holds the hundreds place.
$3.08 \times 10^3 = 3{,}080$

Answer

Written with an exponent, the product is $3.08 \times 10^3$ , and its value is $3{,}080$ .

Here is the idea to keep. When you multiply by a power of ten, every digit shifts one place to the left for each factor of 10. An exponent tells you how many factors there are, so it tells you how many places. The decimal point stays between the ones and the tenths, and any zero you write only fills a place that the shift left empty.

Next, you will go the other way. You will divide a decimal by a power of ten and watch each digit shift to the right.