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Radicals

NCP 503: Work with numerical factors NCP 505: Work with squares and square roots of numbers NCP 507: Work with cubes and cube roots of numbers. Radicals. What is a radical?. Square roots, cube roots, fourth roots, etc are all radicals. They are the opposite of exponents.

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Radicals

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  1. NCP 503: Work with numerical factors NCP 505: Work with squares and square roots of numbers NCP 507: Work with cubes and cube roots of numbers Radicals

  2. What is a radical? • Square roots, cube roots, fourth roots, etc are all radicals. • They are the opposite of exponents. • √4 is asking what number times itself is equal to 4. (Answer is 2) • 3√8 is asking what number times itself “3” times is equal to 8. (Yup…the answer is 2 again.)

  3. Square & Cube Roots Cube Root Square Root √16 3√125 5 ∙ 5 ∙ 5 = 125 So, 3√125 = 5 4 ∙ 4 = 16 So, √16 = 4 Now you know what square and cube roots are, you can figure out the others…fourth root, fifth root, etc.

  4. How to find a cube root! Look for the x√ , to enter 3√125 Enter 3, x√, 125, and EXE.

  5. Try these… • 10 • 8 • 4 • 3 • 3√1000 = • 3√512 = • 3√64 = • 4√81 =

  6. Perfect Squares 2 ∙ 2 = 4, so 4 is a perfect square. Other perfect squares… 9 81 225 16 100 256 25 121 289 36 144 324 49 169 361 64 196 400 The square root of a perfect square is whole number. • √144 = 12

  7. Not So Perfect Squares Finding the square roots of other numbers results in a decimal. WE DO NOT WANT DECIMALS. NO DECIMALS! These will all end up as decimals. Remember: NO DECIMALS! • √50 • √32 • √20

  8. Simplifying Square Roots • √8 = • √4 ∙ √2 = • 2√2 8 is not a perfect square, so we will simplify it! √4 = 2 We can’t simplify √2, so we leave him alone. 8 is made up of 4 ∙ 2. Look! 4 is a perfect square! • √50 = • √25 ∙ √2 = • 5√2

  9. Try these… • = √9 ∙ √3 • = √16 ∙ √2 • = √4 ∙ √5 • = √25 ∙ √3 • = 3√3 • = 4√2 • = 2√5 • = 5√3 √27 √32 √20 √75

  10. Combining Square Roots To combine square roots, combine the coefficients of like square roots. • 4√3 + 5 √3 • = 9√3 They both have √3 in common, so we can add their coefficients. They both have √3 in common, so we can add their coefficients. • 7√2 – 4√2 = • 3√2 Works with subtraction also.

  11. Try these… • 3√5 + 5√5 • 5√7 – 8√7 • -2√3 + 7√3 • 7√11 – 4√11 • √6 + 2√6 • = 8√5 • = -3√7 • = 5√3 • = 3√11 • = 3√6

  12. Combining Square Roots We can combine multiple square roots! • 6√3 + 4√5 – 2√3 + 2√5 • = 4√3 • + 6√5 Combine the √3. Combine the √3. Next, combine the √5. Next, combine the √5. • 4√7 – 5√2 + 3√2 – 2√7 = • 2√7 – 2√2

  13. Try these… • = 3√5 + 3√7 • = 5√2 – 6√3 • = -7√6 + 3√5 • = -6√2 – 5√3 • -2√5 + 3√7 + 5√5 • 5√2 – 8√3 + 2√3 • -4√6 + 2√5 – 3√6 + √5 • √2 – 4√3 – 7√2 – √3

  14. Simplify and Combine √20 + √5 = √4 ∙ √5 + √5 = 3√5 2√5 + √5 = √12 + √27 = √3 ∙ √4 + √9 ∙ √3 = 3√3 = 2√3 + 5√3

  15. Multiplying Radicals When multiplying radicals, you can multiply the two numbers and put the answer under one radical. Simplify! √3 ∙ √2 = √6 √3 ∙ √3 = √9 = 3 √3 ∙ √6 = √18 = √9 ∙ √2 = 3√2

  16. Try This… = 7 √7 ∙ √7 = √49 √3 ∙ √5 = √15 √2 ∙ √6 = √12 = √4 ∙ √3 = 2√3 √15 ∙ √3 = √45 = √9 ∙ √5 = 3√5

  17. Multiplying Radicals • When multiplying radicals, you must multiply the coefficients AND the radicals. THE RADICALS DO NOT HAVE TO BE THE SAME! 2√5 ∙ 3√5 4√2 ∙ 2√8 Let’s see these two examples!

  18. Multiplying Radicals 2√5 ∙ 3√5 1. Multiply the coefficients. 2 ∙ 3 = 6 2. Multiply the radicals. √5 ∙ √5 = √25 3. Solve. 6√25 = 6 ∙ 5 = 30

  19. Multiplying Radicals 4√2 ∙ 2√3 1. Multiply the coefficients. 4 ∙ 2 = 8 2. Multiply the radicals. √2 ∙ √3 = √6 3. Simplify, if possible. 8√6

  20. Try This… 6√35 3√7 ∙ 2√5 = 2√3 ∙ 5√3 = 10√9 = 10 ∙ 3 = 30 4√2 ∙ 3√8 = = 12 ∙ 4 12√16 = 48 2√5 ∙ 3√2 = 6√10

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