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Section 3.4
Division and Exponents
1
Division of Whole Numbers
For any whole numbers r and s, with s ≠ 0, the
quotient of r divided by s, written r ÷ s, is the
whole number k, if it exists, such that r = s × k.
The dividend is r, the divisor is s, and the
quotient is k.
2
Show that 24 ÷ 6 = 4 using both of the
following concepts of division.
Measurement (Subtractive) Concept of Division
How many groups of 6 are in 24?
Sharing (Partitive) Concept of Division
If we have 24 we want to divide into 6 groups,
how many are in each group?
3
What division fact is illustrated?
0
3
6
9
12
4
How could you show 29 ÷ 7 using
base 10 pieces
• Subtractive:
• Partitive:
5
Division Theorem
For any whole numbers a and b with divisor
b ≠ 0, there are whole numbers q (quotient)
and r (remainder) such that
a = bq + r
and 0 ≤ r < b.
Ex: 19 ÷ 5 (19 = a, and 5 = b); we can write:
19 = (5)(quotient) + remainder
Or:
19 = (5)(3) + 4
6
Write the division as multiplication.
1) 102 ÷ 17 = 6
2) 546 ÷ 26 = 21
Write the multiplication as division.
1) 14 x 31 = 434
2) 45 x 7 = 315
7
Method of Equal Quotients
The quotient of two numbers remains the same when
both numbers are divided by the same number.
Use the method of equal quotients to replace
the divisor and the dividend with smaller
numbers. Show the new quotient that
replaces the original quotient. Repeat this
process, if necessary, until you can mentally
calculate the exact quotient.
1) 20 ÷ 10
2) 486 ÷ 18
8
Use compatible numbers to mentally
estimate the quotient.
1) 92 ÷ 14
2) 489 ÷ 47
9
Laws of Exponents
For any number a and all whole numbers
m and n, except for the case where the
base and exponents are both zero,
n
m
n+ m
n
m
n−m
a ×a = a
a ÷a = a
for a ≠ 0
10
Compute the products and quotients.
Leave in exponential form.
10
13
6
9
1) 6 × 6
2) 7 × 7
8
3) 5 ÷ 5
42
3
4) 17 ÷ 17
20
11
Order of Operations
Please Excuse My Dear Aunt Sally
PEMDAS
How does this help us remember the order
of operations?
Evaluate the following expressions.
1) 2 x 8 – 3 x 5
2) (10 + 6) ÷(32 – (4 – 3))
3) 32 ÷ 42 + (5 + 2)2 - 8
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