Simplifying Square RootsFor example: √24 in our calculator gives us 4.89897........ In order to simplify the expression using the calculator method, we would have to round this answer at some point. Let's say we round to the nearest hundredth, and give the answer 4.90. This answer, while very close, is not the exact value. In order to find the exact value, we must actually factor out the greatest perfect square which is the factor of our base, and then take its square root. In this way, we simplify the expression. Some people call this factoring a square root. For example: √24 can be said to be √( 4 x 6 ) We break 24 into 4 x 6 because 4 is the largest perfect square that goes into 24. √( 4 x 6 ) = 2√6 We take the square root of 4, and so 2 comes out to the front of the expression as √4 = 2. The 6 must stay under the radical. Be sure to factor out the largest perfect square for your base. For example: √80 = √( 4 x 20 ) = 2√20 is not correct. 4 is not the largest perfect square that factors into 80. This is evidenced by the fact that the remaining base, 20, can still be factored by the perfect square 4. This is the correct method: √80 = √ ( 16 x 5 ) = 4√5 PRACTICE PROBLEMS 1. Find the exact value of √200 2. Find the exact value of √192 3. Find the exact value of √847 4. Josie worked out her homework problem, and her teacher told her it had been done incorrectly. Can you explain where Josie went wrong? Correctly answer the problem for Josie. √108 = √( 9 x 12 ) = 3√12 ANSWERS 1. √200 = √( 100 x 2 ) = 10√2 2. √192 = √( 64 x 3 ) = 8√3 3. √847 = √( 121 x 7 ) = 11√7 4. Josie did not factor out the largest perfect square that goes into 108. This is how the problem should be worked correctly: √ 108 = √ ( 36 x 3 ) = 6 √ 3 |
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