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Simplifying, Multiplying, & Rationalizing Radicals

Simplifying, Multiplying, & Rationalizing Radicals. Homework: Radical Worksheet. Perfect Squares. 64. 225. 1. 81. 256. 4. 100. 289. 9. 121. 16. 324. 144. 25. 400. …. 169. 36. 196. 49. 625. Simplify. = 2. = 4. = 5. This is a piece of cake!. = 10. = 12.

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Simplifying, Multiplying, & Rationalizing Radicals

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  1. Simplifying, Multiplying, & Rationalizing Radicals Homework: Radical Worksheet

  2. Perfect Squares 64 225 1 81 256 4 100 289 9 121 16 324 144 25 400 … 169 36 196 49 625

  3. Simplify = 2 = 4 = 5 This is a piece of cake! = 10 = 12

  4. Perfect Square Factor * Other Factor Simplify = = = = LEAVE IN RADICAL FORM = = = = = =

  5. Perfect Square Factor * Other Factor Simplify = = = = LEAVE IN RADICAL FORM = = = = = =

  6. Simplify

  7. Simplify

  8. Simplify 4

  9. Simplify OR

  10. Combining Radicals - Addition + To combine radicals: ADD the coefficients of like radicals

  11. Simplify each expression

  12. Simplify each expression: Simplify each radical first and then combine. Not like terms, they can’t be combined Now you have like terms to combine

  13. Simplify each expression: Simplify each radical first and then combine. Not like terms, they can’t be combined Now you have like terms to combine

  14. Multiplying Radicals * • To multiply radicals: • multiply the coefficients • multiply the radicands • simplify the remaining radicals.

  15. Multiply and then simplify

  16. Squaring a Square Root Short cut Short cut

  17. Squaring a Square Root

  18. Dividing Radicals To divide radicals: -divide the coefficients -divide the radicands, if possible -rationalize the denominator so that no radical remains in the denominator

  19. Rationalizing

  20. There is an agreement in mathematics that we don’t leave a radical in the denominator of a fraction.

  21. So how do we change the radical denominator of a fraction? (Without changing the value of the fraction) The same way we change the denominator of any fraction… Multiply by a form of 1. For Example:

  22. By what number can we multiply to change to a rational number? The answer is . . . . . . by itself! Squaring a Square Root gives the Root!

  23. Because we are changing the denominator to a rational number, we call this process rationalizing.

  24. Rationalize the denominator: (Don’t forget to simplify)

  25. Rationalize the denominator: (Don’t forget to simplify) (Don’t forget to simplify)

  26. How do you know when a radical problem is done? • No radicals can be simplified.Example: • There are no fractions in the radical.Example: • There are no radicals in the denominator.Example:

  27. Simplify. Divide the radicals. Simplify.

  28. Simplify. Divide the radicals. Uh oh… There is a radical in the denominator! Whew! It simplified!

  29. Uh oh… Another radical in the denominator! Simplify Whew! It simplified again! I hope they all are like this!

  30. Uh oh… There is a fraction in the radical! Simplify Since the fraction doesn’t reduce, split the radical up. * How do I get rid of the radical in the denominator? Multiply by the “fancy 1” to make the denominator a perfect square!

  31. Fractional form of “1” This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. 42 cannot be simplified, so we are finished.

  32. Simplify fraction Rationalize Denominator

  33. Use any fractional form of “1” that will result in a perfect square Reduce the fraction.

  34. Finding square roots of decimals Find 0.09 Find 1.44 0.09 = 9 ÷ 100 1.44 = 144 ÷ 100 If a number can be made be dividing two square numbers then we can find its square root. For example, = 3 ÷ 10 = 12 ÷ 10 = 0.3 = 1.2

  35. Approximate square roots key on your calculator to find out 2. Use the  If a number cannot be written as a product or quotient of two square numbers then its square root cannot be found exactly. The calculator shows this as 1.414213562 This is an approximation to 9 decimal places. The number of digits after the decimal point is infinite.

  36. Estimating square roots What is 10? Use the key on you calculator to work out the answer.  10 lies between 9 and 16. 10 is closer to 9 than to 16, so 10 will be about 3.2 Therefore, 9 < 10 < 16 So, 3 < 10 < 4 10 = 3.16 (to 2 decimal places.)

  37. Trial and improvement Find 40  Suppose our calculator does not have a key. 40 is closer to 36 than to 49, so 40 will be about 6.3 36 < 40 < 49 So, 6 < 40 < 7 6.32 = 39.69 too small! 6.42 = 40.96 too big!

  38. Trial and improvement 6.332 = 40.0689 too big! 6.322 = 39.9424 too small! Suppose we want the answer to 2 decimal places. 6.3252 = 40.005625 too big! Therefore, 6.32 < 40 < 6.325 40 = 6.32 (to 2 decimal places)

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