Calculating Square Roots – Part 1Slideshow 3, Mr Richard Sasaki, Room 307

Objectives• Recall how to calculate simple

square roots• Calculate square roots to one

decimal place or two significant figures

• Understand cubed roots and other roots

Square RootsSquare roots use the symbol where the number on the right is rooted.So for we’d say “the square root of 4” or just “root 4”.Mathematics is international but there is some disagreement about roots.What is ?

One or Two SolutionsYou need to be understanding that for me (I’m Welsh), = .±3However, in some countries, (at least in Japan), = 3.This will complicate our problems but it’s a very good idea to understand both.

Wales Japan = = = (All the same.)

One or Two Solutions

So, what do we do?In a general case please assume that .If something says “find the positive solution” then of course let .In geometry, makes no sense so almost always .If you don’t know which to do, always ask.Let’s try some questions!

± 4 ±6 ±8±9 ±7 ±1 ±105 110.516−3−13−100−1.5511375

1𝑜𝑟−512𝑜𝑟 02𝑜𝑟−6 We could get +, + or +, - or -, + or -, -.±6±10 ±9

As two negative numbers squared make a positive and two positive numbers make a positive, we can’t square root a negative.

(one solution as it was squared).

Answers

Trial and ErrorHow can we estimate ?We can use a process called Trial and Error.We make a guess and check to see if it is too high or low, then we continue until we get as close to the number as we need to.What number do we square to make 2?12=¿ (Too low) 22=¿ (Too high)1.52=¿ (Too high) 1.42=¿ (Too low)The square root of 2 (to one decimal place) is between 1.4 and 1.5 but closer to 1.4.

√2=±1.4(𝑡𝑜1𝑑 .𝑝)

Trial and ErrorLet’s try to 2 significant figures.

What number do we square to make 34?52=¿ (Too low) 62=¿ (Too high)It’s close to 6…

5.92=¿ (Too high)The square root of 34 (to 2 significant figures) is between 5.8 and 5.9 but closer to 5.8.

√34=±5.8 (𝑡𝑜2𝑠 . 𝑓 )

5.72=¿ (Too low) 5.82=¿ (Too low)

Answers

±1.7

±2 .6

±8 .2

±11

±38

±129

Other RootsWe know square rooting can produce up to two solutions, how about cube rooting?

3√8=¿ (as )¿−2¿−83√−8

We can cube root a negative number but not square root it. (Square rooting it would produce an imaginary number.)How about the fourth root?

4√16=¿ (as )¿−2¿4√16 16

So…For … √𝑥⇒2 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠

3√𝑥⇒1 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛4√𝑥⇒2 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠5√𝑥⇒1 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛6√𝑥⇒2 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠

And so on!ExampleCalculate and . (as but ) (as and )

Answers

351

± 4±60

−2−4−6

2±1 ±10

1±511

40

√√𝑥≡(𝑥 12 )12≡𝑥

14≡ 4√𝑥