example

Multiply:

1. [latex]0\cdot 11[/latex]
2. [latex]\left(42\right)0[/latex]

Solution:

1.	[latex]0\cdot 11[/latex]
The product of any number and zero is zero.	[latex]0[/latex]
2.	[latex]\left(42\right)0[/latex]
Multiplying by zero results in zero.	[latex]0[/latex]

example

Multiply:

1. [latex]\left(11\right)1[/latex]
2. [latex]1\cdot 42[/latex]

Answer: Solution:

1.	[latex]\left(11\right)1[/latex]
The product of any number and one is the number.	[latex]11[/latex]
2.	[latex]1\cdot 42[/latex]
Multiplying by one does not change the value.	[latex]42[/latex]

example

Multiply: [latex-display]8\cdot 7[/latex-display] [latex-display]7\cdot 8[/latex-display]

Answer: Solution:

1.	[latex]8\cdot 7[/latex]
Multiply.	[latex]56[/latex]
2.	[latex]7\cdot 8[/latex]
Multiply.	[latex]56[/latex]

Changing the order of the factors does not change the product.

example

Multiply: [latex]15\cdot 4[/latex]

Answer: Solution

Write the numbers so the digits [latex]5[/latex] and [latex]4[/latex] line up vertically.	[latex]\begin{array}{c}\hfill 15\\ \hfill \underset{\text{_____}}{\times 4}\end{array}[/latex]
Multiply [latex]4[/latex] by the digit in the ones place of [latex]15[/latex]. [latex]4\cdot 5=20[/latex].
Write [latex]0[/latex] in the ones place of the product and carry the [latex]2[/latex] tens.	[latex]\begin{array}{c}\hfill \stackrel{2}{1}5\\ \hfill \underset{\text{_____}}{\times 4}\\ \hfill 0\end{array}[/latex]
Multiply [latex]4[/latex] by the digit in the tens place of [latex]15[/latex]. [latex]4\cdot 1=4[/latex] . Add the [latex]2[/latex] tens we carried. [latex]4+2=6[/latex] .
Write the [latex]6[/latex] in the tens place of the product.	[latex]\begin{array}{c}\hfill \stackrel{2}{1}5\\ \hfill \underset{\text{_____}}{\times 4}\\ \hfill 60\end{array}[/latex]

example

Multiply: [latex]286\cdot 5[/latex]

Answer: Solution

Write the numbers so the digits [latex]5[/latex] and [latex]6[/latex] line up vertically.	[latex]\begin{array}{c}\hfill 286\\ \hfill \underset{\text{_____}}{\times 5}\end{array}[/latex]
Multiply [latex]5[/latex] by the digit in the ones place of [latex]286[/latex]. [latex]5\cdot 6=30[/latex].
Write the [latex]0[/latex] in the ones place of the product and carry the [latex]3[/latex] to the tens place.Multiply [latex]5[/latex] by the digit in the tens place of [latex]286[/latex]. [latex]5\cdot 8=40[/latex] .	[latex]\begin{array}{}\\ \hfill 2\stackrel{3}{8}6\\ \hfill \underset{\text{_____}}{\times 5}\\ \hfill 0\end{array}[/latex]
Add the [latex]3[/latex] tens we carried to get [latex]40+3=43[/latex] . Write the [latex]3[/latex] in the tens place of the product and carry the [latex]4[/latex] to the hundreds place.	[latex]\begin{array}{c}\hfill \stackrel{4}{2}\stackrel{3}{8}6\\ \hfill \underset{\text{_____}}{\times 5}\\ \hfill 30\end{array}[/latex]
Multiply [latex]5[/latex] by the digit in the hundreds place of [latex]286[/latex]. [latex]5\cdot 2=10[/latex]. Add the [latex]4[/latex] hundreds we carried to get [latex]10+4=14[/latex]. Write the [latex]4[/latex] in the hundreds place of the product and the [latex]1[/latex] to the thousands place.	[latex]\begin{array}{c}\hfill \stackrel{4}{2}\stackrel{3}{8}6\\ \hfill \underset{\text{_____}}{\times 5}\\ \hfill 1,430\end{array}[/latex]

example

Multiply: [latex]62\left(87\right)[/latex].

Answer: Solution

Write the numbers so each place lines up vertically.
Start by multiplying 7 by 62. Multiply 7 by the digit in the ones place of 62. [latex]7\cdot 2=14[/latex]. Write the 4 in the ones place of the product and carry the 1 to the tens place.
Multiply 7 by the digit in the tens place of 62. [latex]7\cdot 6=42[/latex]. Add the 1 ten we carried. [latex]42+1=43[/latex] . Write the 3 in the tens place of the product and the 4 in the hundreds place.
The first partial product is[latex]434[/latex].
Now, write a[latex]0[/latex] under the[latex]4[/latex] in the ones place of the next partial product as a placeholder since we now multiply the digit in the tens place of[latex]87[/latex] by[latex]62[/latex]. Multiply[latex]8[/latex] by the digit in the ones place of[latex]62[/latex]. [latex]8\cdot 2=16[/latex]. Write the[latex]6[/latex] in the next place of the product, which is the tens place. Carry the[latex]1[/latex] to the tens place.
Multiply[latex]8[/latex] by[latex]6[/latex], the digit in the tens place of[latex]62[/latex], then add the[latex]1[/latex] ten we carried to get[latex]49[/latex]. Write the[latex]9[/latex] in the hundreds place of the product and the[latex]4[/latex] in the thousands place.
The second partial product is[latex]4960[/latex]. Add the partial products.

The product is [latex]5,394[/latex].

example

Multiply:

1. [latex]47\cdot 10[/latex]
2. [latex]47\cdot 100[/latex]

Answer: Solution

1. [latex]47\cdot 10[/latex]	[latex]\begin{array}{c}\hfill 47\\ \hfill \underset{\text{___}}{\times 10}\\ \hfill 00\\ \hfill \underset{\text{___}}{470}\\ \hfill 470\end{array}[/latex]
2. [latex]47\cdot 100[/latex]	[latex]\begin{array}{c}\hfill 47\\ \hfill \underset{\text{_____}}{\times 100}\\ \hfill 00\\ \hfill \underset{\text{_____}}{\begin{array}{c}\hfill 000\\ \hfill 4700\end{array}}\\ \hfill 4,700\end{array}[/latex]

When we multiplied [latex]47[/latex] times [latex]10[/latex], the product was [latex]470[/latex]. Notice that [latex]10[/latex] has one zero, and we put one zero after [latex]47[/latex] to get the product. When we multiplied [latex]47[/latex] times [latex]100[/latex], the product was [latex]4,700[/latex]. Notice that [latex]100[/latex] has two zeros and we put two zeros after [latex]47[/latex] to get the product. Do you see the pattern? If we multiplied [latex]47[/latex] times [latex]10,000[/latex], which has four zeros, we would put four zeros after [latex]47[/latex] to get the product [latex]470,000[/latex].

example

Multiply: [latex]\left(354\right)\left(438\right)[/latex]

Answer: Solution There are three digits in the factors so there will be [latex]3[/latex] partial products. We do not have to write the [latex]0[/latex] as a placeholder as long as we write each partial product in the correct place.

example

Multiply: [latex]\left(896\right)201[/latex]

Answer: Solution There should be [latex]3[/latex] partial products. The second partial product will be the result of multiplying [latex]896[/latex] by [latex]0[/latex]. Notice that the second partial product of all zeros doesn’t really affect the result. We can place a zero as a placeholder in the tens place and then proceed directly to multiplying by the [latex]2[/latex] in the hundreds place, as shown. Multiply by [latex]10[/latex], but insert only one zero as a placeholder in the tens place. Multiply by [latex]200[/latex], putting the [latex]2[/latex] from the [latex]12[/latex]. [latex]2\cdot 6=12[/latex] in the hundreds place. [latex-display]\begin{array}{}\\ \\ \hfill 896\\ \hfill \underset{\text{_____}}{\times 201}\\ \hfill 896\\ \hfill \underset{\text{__________}}{17920}\\ \hfill 180,096\end{array}[/latex-display]