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Understanding Radicals and Imaginary Numbers: A Comprehensive Guide

Dive into the world of radicals and imaginary numbers with this comprehensive guide! Learn about radical symbols, the principle of square roots, and the steps to simplify various radical expressions. Discover how to find prime factorization and understand the importance of the index in radicals. Plus, explore the fascinating concept of imaginary numbers and complex numbers, including how to work with the imaginary unit 'i'. This resource is perfect for students seeking clarity in mathematics.

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Understanding Radicals and Imaginary Numbers: A Comprehensive Guide

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  1. Dec. 1st HAPPY BIRTHDAY TO: 17th Paul Marron 18thAlissaAndi 24th JJ Mitrokostas 26th Connor Davis

  2. warm-up:graph both

  3. What is a Radical? • A radical symbol is the symbol √ • Usually when we see this, we think square root. That’s because there exists a ninja 2 inside the checkmark part of the radical. It’s called an “index”. • The index tells us how many of some number we need to get the number under the symbol. That number is called the “radicand”. √81 = 9 because we need to multiply 2 nines to get 81 2

  4. Symbols • Radical symbol Radical

  5. Principle Square Root Simplify • √4 is an expression, so we only answer the positive version. • √4 = 2 and is called the “principle square root”.

  6. EXAMPLES √9 = 3 Radicand is 9 Index is 2 “square root” We need two of the same number to multiply by each other to get 9…so the answer is 3. √8 = 2 Radicand is 8 Index is 3 “cube root” We need three of the same number to multiply by each other to get 8…so it’s 2. (2 times 2 times 2 is 8) √16 = 2 “fourth root” 3 4

  7. Square Roots 1) Simplify √16 2) Simplify √25 3) Simplify √100

  8. Square Roots • STEPS: • 1. Find the prime factorization (FACTOR TREE). • 2. Take out any groups based off of the index. • 3. If you take out more than one group, multiple the outside numbers. • 4. If you leave in more than one number, multiply the inside numbers.

  9. Simplify: Factor the 12 12 2 6 2 3 Example 4

  10. Prime Factorization 5) Simplify √48 • I like to divide by 2 as many times as I can, then try 3, 5, and so on (to find prime factorization). • Since the index is 2, we are looking for doubles to take out.

  11. Prime Factorization 7) Simplify √54 • I like to divide by 2 as many times as I can, then try 3, 5, and so on (to find prime factorization). • Since the index is 2, we are looking for doubles to take out.

  12. Prime Factorization 8) Simplify √372 • I like to divide by 2 as many times as I can, then try 3, 5, and so on (to find prime factorization). • Since the index is 2, we are looking for doubles to take out.

  13. Another Option 9) Simplify √75 • Another option is to look for perfect squares to take out, such as 4, 9, 16, 25, 100, etc. • Since the index is 2, we are looking for doubles to take out.

  14. Another Option 10) Simplify √200 • Another option is to look for perfect squares to take out, such as 4, 9, 16, 25, 100, etc. • Since the index is 2, we are looking for doubles to take out.

  15. Another Option 11) Simplify √32 • Another option is to look for perfect squares to take out, such as 4, 9, 16, 25, 100, etc. • Since the index is 2, we are looking for doubles to take out.

  16. IMAGINARY NUMBERS…………. • Not all equations have a real-number solution. • Sometimes you could get something like: x2 = -16 • To be able to solve these types of problems the imaginary number was created. √

  17. Complex Numbers Real Numbers Imaginary Numbers RationalIrrational

  18. The Imaginary Unit

  19. Complex Numbers The set of all numbers that can be written in the format: a + bi ; “a” is the real number part “bi’ is the imaginary part

  20. bi 4i 2i + 7i = 9i 8 + i 2i+ 5i= 7i Take the i out first

  21. HOMEWORK • Worksheet

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