Multiplication of radicals
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This document discusses the three cases of multiplying radicals: 1) Same indices - the indices are multiplied but the radicands are multiplied, e.g. 3√4 * 4√4 = 12√4 2) Different indices but same radicand - the indices are added but the radicand is multiplied, e.g. 5√2 * 5√3 = 5√6 3) Totally different indices and radicands - the indices and radicands are multiplied separately, e.g. 4√3 * 5√2 = 20√6 Examples and practice problems are provided to illustrate each case.
Multiplication of radicals
- 3. Add the following radicals • 3 4 , 48 4 • 5 2 , 6 2 , 20 2 , 294 2 • 𝑥43 + 𝑥73
- 5. Three cases of multiplications of radicals • Same indices • Different indices but same radicand • Totally different in indices and radicands
- 6. First case: Same indices • 𝑥𝑛 ∙ 𝑦𝑛 = 𝑥𝑦𝑛
- 7. Examples • 3 4 ∙ 4 4 • 3 ∙ 4 4 • 12 4 • 10 5 ∙ 2 5 • 10 ∙ 2 5 • 20 5
- 8. Boardwork: • 10 5 ∙ 20 5 • 16 6 ∙ 4 6 • 12800 7 ∙ 1000 7 ∙ 100 7
- 9. Practice exercise • Simplify the following radicals: • ( 6 4 ∙ 10 4 )( 9 4 ∙ 10 4 ) • 4 3 ∙ 16 3 2 • Show that the product of the roots of a quadratic equation is 𝑐 𝑎 .
- 10. Second case: Different indices same radicand • 𝑛 𝑎𝑥 ∙ 𝑛 𝑎𝑦 • 𝑛 𝑎 𝑥 ∙ 𝑛 𝑎 𝑦 • 𝑛 𝑎𝑦 𝑥𝑦 ∙ 𝑛 𝑎𝑥 𝑥𝑦 • 𝑛 𝑎𝑦+𝑎𝑥 𝑥𝑦 • 𝑛 𝑎𝑦+𝑎𝑥 𝑥𝑦
- 11. Example • 5 2 ∙ 5 3 • 5 1 2 ∗ 5 1 3 • 5 3 6 ∗ 5 2 6 • 5 5 6 • 3125 6
- 12. Boardwork • Simplify the following expressions: • 8 4 ∗ 8 5 • 𝑥710 ∗ 𝑥720 • 𝑦𝑥 ∗ 𝑦𝑧
- 13. Practice Exercise • Simplify the following to simplest radical form. Show the step-by-step solution
- 14. • 1. 6 3 ∗ 6 2 ∗ 6 4 • 2. ( 𝑎2 + 𝑏2)( 𝑎2 + 𝑏23 ) • 3. 72 ∗ 16 3
- 15. Case 3: Totally Different • 𝑛 𝑎𝑥 ∙ 𝑚 𝑏 𝑦 • 𝑛 𝑎 𝑥 ∙ 𝑚 𝑏 𝑦 • 𝑛 𝑎𝑦 𝑥𝑦 ∙ 𝑚 𝑏𝑥 𝑥𝑦 • 𝑛 𝑎𝑦 ∗ 𝑚 𝑏𝑥 𝑥𝑦
- 16. Example • 4 3 ∗ 5 2 • 4 1 3 ∗ 5 1 2 • 4 2 6 ∗ 5 3 6 • 42 ∗ 536 • 2000 6
- 17. Boardwork • Simplify the following: • 5 3 ∗ 2 2 ∗ 7 4 • 𝑥10 ∗ 𝑥92015 (kudos to who will answer this)
- 18. Practice exercise • Simplify the following expressions: • 𝑐2 ∗ 3𝑐 3 • ( 𝑎2 + 𝑏2)( 𝑎2 − 𝑏23 )