Appendix A

A:01C Long Multiplication and Long Division
Multiplication
Division
1 Estimate first. 1 Estimate first.
2 Multiply by the ones digit. 2 Use either the contracted form method or the
3 Multiply by the tens digit. preferred multiples method.
4 Add the two answers. 3 Write the answer using a remainder or fraction.
Example
Extended form
531 ×
48
4 248 ← 8 × 531
21 240 ← 40 × 531
25 488
A:01D Powers of Numbers
five squared
5 2 = 5 × 5
= 25
five cubed
5 3 = 5 × 5 × 5
= 125
Example (long division)
Contracted form
28 r 10
31) 878
− 62↓
258
248
10
Answer: 28 remainder 10 or 28 10 -----
31
Preferred multiples
28 r 10
31) 878
− 620 20
258
− 248 8
10 28
■ The small raised number (the index) tells how
many times that number is used in the product.
worked examples
1 Write 2 5 in expanded form and write its basic numeral.
2 Write 10 × 10 × 10 × 10 as a power of ten.
3 What is the basic numeral of 8 × 10 4 ?
Solutions
1 2 5 2 10 × 10 × 10 × 10 3 8 × 10 4
= 2 × 2 × 2 × 2 × 2 = 10 4 = 8 × (10 × 10 × 10 × 10)
= 32 = 8 × 10 000
= 80 000
A:01E Expanded Notation Using Powers
Ten-thousands
10 000
Thousands
1000
Hundreds
100
Tens
10
Units
1
10 × 10 × 10 × 10 10 × 10 × 10 10 × 10 10 1
10 4 10 3 10 2 10 1 1
6 9 8 7 5
■ 6 ten-thousands
9 thousands
8 hundreds
7 tens
5 units
69 875
69 875 = (6 × 10 000) + (9 × 1000) + (8 × 100) + (7 × 10) + (5 × 1)
= (6 × 10 4 ) + (9 × 10 3 ) + (8 × 10 2 ) + (7 × 10) + (5 × 1)
APPENDIX A 3