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 themselves in a pattern. This pattern continues infinitely.0.3333333…=¿0. 3̇We know how some of these numbers convert to fractions.

¿13But some numbers we may need to do calculations to write them as fractions.

Recurring Decimal NumbersFor a number like , just one digit repeats (the 7). We need to change this number to get rid of the recurring symbol.If we multiply this number by 10 and then subtract the original (multiply by 9), the recurring digit will disappear.ExampleWrite as a fraction.Let .

10 𝑥=¿7 . 7̇9 𝑥=¿7

∴𝑥=79

Recurring Decimal NumbersIf we have more than a group of digits with recurring symbols (eg: ), those digits repeat in that sequence. (eg: 0.374374374…)ExampleWriteas a fraction.Let .

To remove recurrence, what should we multiply it by?

1000 𝑥=374. 3̇ 7̇ 4̇999 𝑥=374

∴𝑥=374999

Recurring Decimal NumbersThe multiplication process differs with numbers where the first recurring digit isn’t in the tenths position (eg:

Note: The 4 doesn’t recur.Exampl

eWrite as a fraction.Let .10 𝑥=104. 2̇3̇

1000 𝑥=10423. 2̇3̇990 𝑥=10319

∴𝑥=10319990

Answers

29269173

11897339899

10433

142099

41333

445

389909215

12679060774959589135

Surd LawsLast lesson, we learned how to simplify surds. The primary rule we learned was:

, where .

This is, of course useful for multiplying surds.ExampleSimplify .

√6×√3=¿√18=¿√2√9=¿3√2So typically, we combine the surds and then simplify it as one expression. This is normally the easiest method.

Surd LawsOf course, surds are often in the form where .ExampleSimplify .Let’s separate it into four chunks and then combine them.2×√5×4×√12=¿8×√60=¿8 √60Lastly, let’s simplify .

8 √60=¿8 √4√15=¿8 ∙2√15=¿16√15So you may use the fact…

for .

Answers

√6 2√10 4√15

6 √35 16√5 10√147√6 ±6 2√285

28√105 96 √30 168√39

±280175 180√19

Dividing RootsHow do we divide square roots?Let’s consider two roots, and where .

If we square both sides, we get…

𝑥2=( √𝑎√𝑏 )

2

𝑥=√ 𝑎𝑏¿ √𝑎√𝑏∙ √𝑎√𝑏

If we square root both sides, we get… , where .

¿(√𝑎)2

(√𝑏 )2¿𝑎𝑏

Dividing RootsDividing surds is often similar to multiplying them.ExampleCalculate and simplify .

4√8÷3√2=¿4 √83 √2

=¿4 √43

=¿4 ∙ ±23

=¿±83

Calculate and simplify .

12√1083 √3

=¿4√36=¿4 ∙ ±6=¿±24

If the numbers are kind to you, the question is easy.

Dividing RootsWhen we divide, the number being rooted may become a fraction.ExampleCalculate and simplify .

√2÷√3=¿√2√3

We always need an integer as the denominator.

√2√3

=¿√2∙√3√3 ∙√3

=¿√63

If we square the surd denominator, it will become an integer.

When you write a fraction, it must be in the form

where .

Answers - Easy

±2 ±2 2√102√67

±152 ±24

±5 ± 74 2√2

5√104 ±

965

55√24

4 √313 14 √63√2

Answers - Hard

√155 √13 √6

2

√22 ±

43

3√142

√1414

√3311

3√38

9√28

99√24

7√10426

√21022

55√21063 ±

310