Problem 2 Simplifying to Combine Like Radicals

What is the simplified form of 5 3 − 12 ? 5 square root of 3 minus square root of 12 question mark

5 3 − 12 = 5 3 − 4 · 3 4  is a perfect-square factor of  12.   = 5 3 − 4 · 3 Multiplication Property of Square Roots   = 5 3 − 2 3 Simplify   4 .   = ( 5 − 2 ) 3 Use the Distributive Property to combine like radicals .   = 3 3 Simplify . table with 5 rows and 3 columns , row1 column 1 , 5 square root of 3 minus square root of 12 , column 2 equals 5 square root of 3 minus , square root of 4 middle dot 3 end root , column 3 4 . isaperfectminussquarefactorof . 12. , row2 column 1 , , column 2 equals 5 square root of 3 minus square root of 4 middle dot square root of 3 , column 3 cap multiplicationcap propertyofcap squarecap roots , row3 column 1 , , column 2 equals 5 square root of 3 minus 2 square root of 3 , column 3 cap simplify , square root of 4 . , row4 column 1 , , column 2 equals . open , 5 minus 2 , close . square root of 3 , column 3 cap usethecap distributivecap propertytocombinelikeradicals . . , row5 column 1 , , column 2 equals 3 square root of 3 , column 3 cap simplify , . , end table

✓ Got It?

2. What is the simplified form of each expression in parts (a) and (b)?
   1. 4 7 + 2 28 4 square root of 7 plus 2 square root of 28
   2. 5 32 − 4 18 5 square root of 32 minus 4 square root of 18
   3. Reasoning Can you combine two unlike radicals when the radicands have no common factors other than 1? Explain.

Problem 3 Multiplying Radical Expressions

What is the simplified form of each expression?

1. 10 ( 6 + 3 ) square root of 10 . open , square root of 6 plus 3 , close

   10 ( 6 + 3 ) = ( 10 · 6 ) + ( 10 · 3 ) Distributive Property   = 60 + 3 10 Multiplication Property of Square Roots   = 4 · 15 + 3 10 4  is a perfect-square factor of  60.   = 2 15 + 3 10 Simplify   4 . table with 4 rows and 3 columns , row1 column 1 , square root of 10 . open , square root of 6 plus 3 , close , column 2 equals . open . square root of 10 middle dot square root of 6 . close . plus . open , square root of 10 middle dot 3 , close , column 3 cap distributivecap property , row2 column 1 , , column 2 equals square root of 60 plus 3 square root of 10 , column 3 cap multiplicationcap propertyofcap squarecap roots , row3 column 1 , , column 2 equals square root of 4 middle dot square root of 15 plus 3 square root of 10 , column 3 4 . isaperfectminussquarefactorof . 60. , row4 column 1 , , column 2 equals 2 square root of 15 plus 3 square root of 10 , column 3 cap simplify , square root of 4 . , end table

2. ( 6 − 2 3 ) ( 6 + 3 ) open . square root of 6 minus 2 square root of 3 . close . open , square root of 6 plus square root of 3 , close

   ( 6 − 2 3 ) ( 6 + 3 ) = 36 + 18 − 2 18 − 2 9 Use FOIL .   = 6 − 18 − 2 ( 3 ) Combine like radicals and simplify .   = 6 − 9 · 2 − 6 9  is a perfect-square factor of  18.   = − 3 2 Simplify . table with 4 rows and 3 columns , row1 column 1 , open . square root of 6 minus 2 square root of 3 . close . open , square root of 6 plus square root of 3 , close , column 2 equals square root of 36 plus square root of 18 minus 2 square root of 18 minus 2 square root of 9 , column 3 cap usecap fcap ocap icap l , . , row2 column 1 , , column 2 equals 6 minus square root of 18 minus 2 , open 3 close , column 3 table with 2 rows and 1 column , row1 column 1 , cap combinelikeradicals , row2 column 1 , andsimplify . . , end table , row3 column 1 , , column 2 equals 6 minus square root of 9 middle dot square root of 2 minus 6 , column 3 table with 2 rows and 1 column , row1 column 1 , 9 . isaperfectminussquare , row2 column 1 , factorof . 18. , end table , row4 column 1 , , column 2 equals negative 3 square root of 2 , column 3 cap simplify , . , end table

✓ Got It?

3. What is the simplified form of each expression?
   1. 2 ( 6 + 5 ) square root of 2 . open , square root of 6 plus 5 , close
   2. ( 11 − 2 ) 2 open , square root of 11 minus 2 , close squared
   3. ( 6 − 2 3 ) ( 4 3 + 3 6 ) open . square root of 6 minus 2 square root of 3 . close . open . 4 square root of 3 plus 3 square root of 6 . close