1. Calculating Square Roots – Part 2
Slideshow 4, Mathematics
Mr Richard Sasaki

2. Objectives
Be able to calculate square roots for fractions and decimals
Be able to compare square roots

3. Square Roots - Rules Let us consider both roots for 𝑥 2 , . ±𝑥
When we multiply and divide, we consider only positive roots or negative roots. We don’t need to worry about one of each.
𝑥 2 ∙ 𝑦 2 =
±𝑥∙±𝑦=
±𝑥𝑦
This is because…
𝑥∙𝑦= 𝑥𝑦
−𝑥∙𝑦=
−𝑥𝑦 𝑥𝑦
𝑥∙−𝑦=
−𝑥𝑦
−𝑥∙−𝑦=
These four combinations result in just two possible answers.

4. Square Roots - Review
Previously, we found the positive root of 2.25 is 1.5. If it’s hard to calculate this mentally, an easier way may be to convert it to an improper fraction.
9 4 = =
± 3 2
2.25 =
Here, we used the fact that 𝑎 𝑏 = 𝑎 𝑏 .
We can use this at any time. It can make finding roots of numbers with fractions and decimals easier.

5. Square Roots - Decimal Example
Find the positive root of and write your answer as a fraction. = = =
4 5
0.64 =
Find the positive root of = = = =
18 5
12.96 =

6. Answers - Easy ± 5 6 ± 7 8 ± 3 10 ± 3 4 ± 2 7 ± 1 11 ± 8 9 ± 15 13
±0.9
±1.1
±0.5
±0.6
±1.2
±0.4
±1.3
±0.1
3 4
1 8
2 3
5 4
2 5
6 7

7. Answers - Medium ± 8 3 ± 4 9 ± 7 10 ± 4 11 ± 5 7 ± 3 8 1.6 1.5 2.2 2.8
1.8
2.6
2.4
2.7
± 6 7
± 3 4
± 2 11
± 2 5

8. Answers - Hard
± 7 8
± 9 4
± 10 11
± 5 12
4.2
7.6
9.8
7.5
4.6
6.6
9.5
3.2
8.7
8.3
8.1
5.4

9. Comparing Roots
As we know, 4<5. This implies that for positive roots, 16 < 25 .
In the same way…
If 𝑥<𝑦, then positive roots 𝑥 < 𝑦 for all 𝑥, 𝑦∈ ℝ 0 + .
All positive real numbers and zero.
Remember, 𝑥 and 𝑦 must be positive! (Roots of negative numbers produce imaginary ones!)

10. Comparing Roots Examples
Place an inequality sign between the following. Consider positive roots only.
<
>
9 15 = 3 5
>
This is true because 3 4 > 3 5 .

11. <
>
>
>
>
<
>
>
<
>
<
>
> =
>