Algebra 1 (W1-S1) Lesson - Cardiff University

Cardiff University International Study Centre Week 1 Session 1 Algebra 1 Learning Objectives To be able to multiply, divide, add and subtract surds...

Cardiff University International Study Centre Week 1 Session 1 Algebra 1 Learning Objectives To be able to multiply, divide, add and subtract surds To be able to simplify surds To be able to use the laws of indices  Section 1 Surds Multiplying surds Dividing surds Simplifying surds Adding surds Subtracting surds 2 Key points Multiplying surds 𝑎 × 𝑏 = 𝑎𝑏 Dividing surds 𝑎 𝑎 ÷ 𝑏= 𝑏 Simplifying surds We can take out square numbers and find their root 4𝑎 = 4 × 𝑎 = 2 𝑎 Adding and subtracting surds If we have several of the same surd, we can add or subtract them 𝑏+2 𝑏 =3 𝑏 3 Example 1 – Multiplying surds Multiply the following surds. 2× 3 3 × 10 5× 2 2 5× 7 12 × 3 3 3 6×4 2 4 Example 2 – Dividing surds Divide the following surds. 10 ÷ 2 12 ÷ 6 3÷ 7 2 30 ÷ 6 10 ÷ 3 2 6 14 ÷ 2 2 5 Example 3 – Simplifying surds Simplify the following surds. 25 12 200 2 32 3 156 5 2 10 6 Example 4 – Adding and subtracting surds Add and subtract the following surds. 5+3 5 2 2− 2 20 + 45 2 32 − 4 2 48 + 6 12 180 − 20 − 2 5 7 Exercise Compute and simplify the following. 1) 3 5 × 3 3 6) 5 2 × 5 6 2) 4 21 ÷ 2 3 7) 4 3 × 2 6 3) 32 − 8 8) 6 42 ÷ 2 7 4) 20 30 ÷ 5 6 9) 2 3 × 2 5 5) 507 − 48 − 75 10) 320 − 20 − 20 Calculator note: Only use a calculator to help with multiplying or dividing numbers! You need to do the work by hand to get marks on the exam. 8 Answers 1) 3 5 × 3 3 ⟶ 9 15 6) 5 2 × 5 6 ⟶ 50 3 2) 4 21 ÷ 2 3 ⟶2 7 7) 4 3 × 2 6 ⟶ 24 2 3) 32 − 8 ⟶2 2 8) 6 42 ÷ 2 7 ⟶3 6 4) 20 30 ÷ 5 6 ⟶4 5 9) 2 3 × 2 5 ⟶ 4 15 5) 507 − 48 − 75 ⟶4 3 10) 320 − 20 − 20 ⟶4 5 9  Section 2 Indices Multiplying indices Dividing indices Powers of indices Negative indices Fractional indices 10 Key points Multiplying indices 𝑥 ×𝑥 =𝑥 Dividing indices 𝑥 ÷𝑥 =𝑥 Powers of indices 𝑥 =𝑥 Negative indices 1 𝑥 = 𝑥 Fractional indices 𝑥 = 𝑥 11 Example 1 – Multiplying indices Multiply the following indices. 𝑥 ×𝑥 3𝑥 × 5𝑥 𝑥 ×𝑥×𝑥 𝑥 ×𝑥 𝑥 ×𝑦 2𝑥 𝑦 × 𝑥 𝑦 12 Example 2 – Dividing indices Divide the following indices. 𝑥 ÷𝑥 𝑥 ÷𝑥 15𝑥 ÷ 3𝑥 𝑥 ÷𝑥 2𝑥 ÷ 3𝑦 2𝑥 𝑦 ÷ 𝑥𝑦 13 Example 3 – Powers of indices Simplify the following expressions. 𝑥 2 2𝑦 7𝑥 2𝑥𝑦 3 4𝑎𝑏 14 Example 4 – Power of zero Simplify the following. 𝑥 4 3𝑥 (3𝑥) 15 Example 5 – Negative indices These expressions are written in “index form”. Write them in ordinary form. 𝑥 𝑥 4 5𝑥 5𝑥 1 𝑥 7 16 Example 6 – Fractional indices These expressions are written in “index form”. Write them in ordinary form. 𝑥 𝑥 27 16 4𝑥 17 Exercise Match the ordinary form expressions (numbers) with their index form expressions (letters). 1 1 1 𝑥 11 × A 𝑥 K 𝑥 𝑥 𝑥 1 2 12 𝑥 B 1 L 𝑥 𝑥 𝑥 1 2 3 𝑥 13 𝑥 1 C 𝑥 M 𝑥 2 1 4 𝑥 14 𝑥 D 𝑥 N 𝑥 1 1 5 15 𝑥 𝑥 E 𝑥 O 2𝑥 1 6 16 𝑥 ×𝑥 1 2𝑥 F 𝑥 P 𝑥 2 4 2 7 17 𝑥 𝑥 G 𝑥 Q 𝑥 1 8 𝑥 𝑥 18 H 𝑥 R 4𝑥 𝑥 𝑥 𝑥 9 19 I 2𝑥 S 2𝑥 𝑥 4 1 10 20 4𝑥 J 𝑥 T 𝑥 𝑥 18 Answers 1 1 1P 𝑥 11 Q × A 𝑥 K 𝑥 𝑥 𝑥 1 2E 12 R 𝑥 1 𝑥 B 𝑥 L 𝑥 1 2 3K 𝑥 13 T 𝑥 1 C 𝑥 M 𝑥 2 1 4J 𝑥 14 D 𝑥 D 𝑥 N 𝑥 1 1 5M 15 G 𝑥 𝑥 E 𝑥 O 2𝑥 1 6B 16 L 𝑥 ×𝑥 1 2𝑥 F 𝑥 P 𝑥 2 4 2 7S 17 O 𝑥 𝑥 G 𝑥 Q 𝑥 1 8N 𝑥 𝑥 18 C H 𝑥 R 4𝑥 𝑥 𝑥 𝑥 9A 19 F I 2𝑥 S 2𝑥 𝑥 4 1 10 H 20 I 4𝑥 J 𝑥 T 𝑥 𝑥 19