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

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

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.

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 = 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.

Square Roots - Decimal Example Find the positive root of 0.64 and write your answer as a fraction. 64 100 = 16 25 = 16 25 = 4 5 0.64 = Find the positive root of 12.96 . 1296 100 = 648 50 = 324 25 = 324 25 = 18 5 12.96 =

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

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

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

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!)

Comparing Roots Examples Place an inequality sign between the following. Consider positive roots only. 8 15 4 13 < > 3 4 9 15 9 15 = 3 5 > This is true because 3 4 > 3 5 .

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