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1 Section 9 2A: Simplifying Radical Expressions Rational Numbers A Rational Number is any number that that expressed as a whole number a fraction a decimal that ends a decimal that repeats In the last section every number under the square root symbol was a perfect square. The square root of a perfect square replaced with a rational number. 9 replaced with a replaced with a 2 replaced with a 6 5 replaced with a 4 3 or Irrational numbers Irrational Numbers are decimal numbers whose digits go on and on without ending and the digits never repeat in any pattern. Irrational Numbers are numbers that CANNOT BE EXPRESSED as a whole number, a fraction, a decimal that ends or a decimal that repeats. The square root of a number that is not a perfect square is an irrational number. You cannot write a decimal that does not end or repeat. This means that the square root of numbers that are not perfect squares cannot be written without a radical sign. is an irrational number. It cannot be replaced with a whole number, a decimal or fraction. 21 is an irrational number. It cannot be replaced with a whole number, a decimal or fraction is an irrational number. It cannot be replaced with a whole number, a decimal or fraction. 13 Math 100 Section 9 2A Page Eitel

3 Reducing a Square Root Expression If the radicand is an irrational number that has a factor that is a perfect square then the square root can be reduced to a rational number outside the radical sign times the square root of an irrational number smaller than the original radicand you started with. This process is called reducing or simplifying the square root. Reducing a square root requires the use of the Multiplication Rule for Square Roots. The Multiplication Rule for Square Roots C = A B = A B The Multiplication Rule for Square Roots allows you to factor the number under the square root and write the positive factors as a product with each factor under a separate square root. 15 = 3 5 = = 9 = 9 Simplifying Square Roots using PERFECT SQUARE FACTORS This process requires that you find the largest perfect square that is a factor of the original radicand. Use the list of perfect squares below as a guide for finding the largest perfect square that is a factor of the original radicand for radicands whose value is from 1 to Look for the largest perfect square in the list above that is a factor of the radicand. Factor the radicand and put each factor under a separate square root. Reduce the square root of the perfect square and multiply that number times the remaining square root. Example 1 Example 2 Simplify: 63 Simplify: 12 Factor using a 9 = 9 = 9 Factor using a 4 = 4 3 = 4 3 = 3 = 2 3 Math 100 Section 9 2A Page Eitel

4 Example 3 Example 4 Simplify: 44 Simplify: 125 Factor using a 4 = 4 11 = 4 11 Factor using a 25 = 25 5 = 25 5 = 2 11 = 5 5 Example 5 Example 6 Simplify: 18 Simplify: 48 Factor using a 9 = 9 2 = 9 2 Factor using a 16 = 16 3 = 16 3 = 3 2 = 4 3 Example 5 Note: To use this technique you must factor 18 as 9 2 and not as 3 6 Example 6 Note: To use this technique you must factor 48 as 16 3 and not as 4 12 Example Example 8 Simplify: 80 Factor using a 36 = 36 2 = 36 2 Factor using a 16 = 16 5 = 16 5 Example Note: To use this technique you must factor 2 as 36 2 and not as 9 8 Math 100 Section 9 2A Page Eitel

5 Simplifying Square Root Radicals using PAIRS OF FACTORS If you can find the largest perfect square factor of the radicand then reducing the radical expression is a short process. Many students cannot find the largest perfect square factor or they do not want to take the extended time this may take. There is an alternate approach that is favored by many students. A pair (two) of the same factors under a square root form a perfect square. This means that if you have a pair of the same factors under a square root they reduced to a rational number. 2 2 = 4 = 2 a pair of 2's under a square root reduce to the whole number = 9 = 3 a pair of 3's under a square root reduce to the whole number = 25 = 5 a pair of 5's under a square root reduce to the whole number 5 This fact allows us to use the Multiplication Rule for Square Roots to completely factor a radicand into its many factors and then take out any pairs of the same factors. Completely factor the radicand and take out any pairs of the same factors. Example 1 Example 2 Example 3 Simplify: 24 completly factor 24 Simplify: 54 completly factor 54 Simplify: 48 completly factor 48 = put pairs of the same factor under thier own square root = put pairs of the same factor under thier own square root = put pairs of the same factor under thier own square root = the pair of 2's can reduced = 2 6 = the pair of 3's can reduced = 3 6 = each pair of 2's can reduced = = 4 3 Example 4 Example 5 Example 6 Simplify: 18 = 6 3 = = = 3 2 Simplify: 60 = = = 2 15 Simplify: 50 = 5 10 = = = 5 2 Math 100 Section 9 2A Page Eitel

6 Example Example 8 Example 9 = = = = = = = = = = = 2 30 Simplifying Square Roots using both Pairs of Factors and Perfect Squares Factor the radicand. If a factor is a perfect square do not factor it further. Take out all the pairs and Perfect Squares: Example 10 Example 11 Example 12 = = = Simplify: 80 = 4 20 = = = 4 5 Simplify: 120 = 40 3 = = = = 2 30 Example 13 Example 14 Example 15 = = = Simplify: 80 = 4 20 = = = = Simplify: 180 = = = = 6 5 Math 100 Section 9 2A Page Eitel

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