Download Multiplying and Dividing Surds

Multiplying and Dividing
Surds
Slideshow 7, Mr Richard Sasaki, Room 307
Objectives
• Learn (or review) how to write recurring
decimal numbers as fractions
• Learn how to multiply and simplify numbers
in surd form
• Learn how to divide and simplify numbers in
surd form
Recurring Decimal Numbers
A recurring decimal number is a number where
decimal digits repeat in a pattern infinitely.
0.3333333 … = 0. 3 = 1 3
We know how simple cases convert to fractions.
But some we need to calculate.
For a number like 0. 7, just one digit repeats. If we
multiply this by 10 and then subtract the original
(multiply by 9), the recurring digit will disappear.
Example
Write 0. 7 as a fraction.
Let 𝑥 = 0. 7.
7
∴𝑥=
10𝑥 = 7. 7
9
9𝑥 = 7
Recurring Decimal Numbers
If we have more than a group of digits with
recurring symbols (eg: 0. 374), those digits repeat
in that sequence. (eg: 0.374374374 …)
Example
Write 0. 374 as a fraction.
Let 𝑥 = 0. 374.
1000𝑥 = 374. 374
999𝑥 = 374
Think about what we should
multiply the original number by.
374
∴𝑥=
999
Recurring Decimal Numbers
The multiplication process differs with numbers
where the first recurring digit isn’t in the tenths
position (eg: 10.423 = 10.423232323 … )
Note: The 4 doesn’t recur.
Example
Write10.423 as a fraction.
Let 𝑥 = 10.423.
10319
10𝑥 = 104. 23
∴𝑥=
990
1000𝑥 = 10423. 23
990𝑥 = 10319
Answers
2
9
26
9
17
3
118
9
7
33
98
99
104
33
1420
99
41
333
4
45
389
90
92
15
1267
90
6077
495
1
7
Surd Laws
Last lesson, we learned how to simplify surds.
The primary rule we learned was:
∴ 𝑎 × 𝑏 ≡ 𝑎 × 𝑏, where 𝑎, 𝑏 ∈ ℝ.
This is, of course useful for multiplying surds.
Example
Simplify 6 × 3.
6 × 3 = 18 = 2 9 = 3 2
So typically, we combine the surds and then
simplify it as one expression. This is normally the
easiest method.
Surd Laws
Of course, surds are often in the form 𝑎 𝑏 where
𝑎 > 1.
Example
Simplify 2 5 × 4 12.
Let’s separate it into four chunks and then
combine them.
2 × 5 × 4 × 12 = 8 × 60 = 8 60
Lastly, let’s simplify 8 60.
8 60 = 8 4 15 = 8 ∙ 2 15 = 16 15
So you may use the fact…
𝑎 𝑐 × 𝑏 𝑑 = 𝑎𝑏 𝑐𝑑 for 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℤ.
Answers
6
2 10
6 35
16 5
7 6
6
28 105
280
4 15
10 14
2 285
96 30
168 39
175
180 19
Dividing Roots
How do we divide square roots?
Let’s consider two roots, 𝑎 and 𝑏 where 𝑥 =
If we square both sides, we get…
𝑎
𝑥2 =
2
=
𝑏
𝑎
𝑏
∙
𝑎
𝑏
=
𝑎
𝑏
2
2
𝑎
=
𝑏
If we square root both sides, we get… 𝑥 =
∴
𝑎
𝑏
≡
𝑎
,
𝑏
where 𝑎, 𝑏 ∈ ℝ.
𝑎
𝑏
𝑎
.
𝑏
Dividing Roots
Dividing surds is usually similar to multiplying
them.
Example
State the positive root for 4 8 ÷ 3 2.
4 8 4 4 4∙2 8
4 8÷3 2=
=
=
=
3
3
3
3 2
State both roots for
12 108
3 3
12 108
.
3 3
= 4 36 = 4 ∙ ±6 = ±24
Dividing Roots
When dividing, we must make sure the
denominator is an integer.
Example
Calculate and simplify 2 ÷ 3.
6
2∙ 3
2
2÷ 3=
=
=
3
3∙ 3
3
When you write a fraction, it must
be in the form
𝑎 𝑏
𝑐
where 𝑎, 𝑏, 𝑐 ∈ ℤ.
If we square the surd denominator, it will
become an integer.
2
2
2 10
2 6
7
15
2
24
5
7
4
2 2
96
5
55 2
4
14 6
3 2
5 10
4
4 31
3
15
5
2
2
4
3
6
2
3 14
2
14
14
9 2
8
33
11
99 2
4
3 3
8
7 1042
6
55 210
63
3
10
210
22
13