Practice and Problem-Solving Exercises

See Problem 1.

A Practice

Simplify if possible.

10. 5 6 + 6 5 square root of 6 plus square root of 6
11. 6 3 3 − 2 3 3 6 , cube root of 3 , , minus 2 , cube root of 3 ,
12. 4 3 + 4 3 3 4 square root of 3 plus 4 , cube root of 3 ,
13. 3 x − 5 x 3 square root of x minus 5 square root of x
14. 14 x + 3 y 14 square root of x plus 3 square root of y
15. 7 x 2 3 − 2 x 2 3 7 , cube root of x squared end root , , minus 2 , cube root of x squared end root ,

    See Problem 2.

16. The design of a garden path uses stone pieces shaped as squares with a side length of 15 in. Find the length of the path.
    Image Long Description

See Problem 3.

Simplify.

17. 6 18 + 3 50 6 square root of 18 plus 3 square root of 50
18. 14 20 − 3 125 14 , square root of 20 minus 3 , square root of 125
19. 18 + 32 square root of 18 plus square root of 32
20. 54 3 + 16 3 cube root of 54 , , plus , cube root of 16 ,
21. 3 81 3 − 2 54 3 3 , cube root of 81 , , minus 2 , cube root of 54 ,
22. 32 4 + 48 4 the fourth , root of 32 , , plus , the fourth , root of 48 ,

See Problem 4.

Multiply.

23. ( 3 + 5 ) ( 1 + 5 ) open , 3 plus square root of 5 , close . open , 1 plus square root of 5 , close
24. ( 2 + 7 ) ( 1 + 3 7 ) open , 2 plus square root of 7 , close . open . 1 plus 3 square root of 7 . close
25. ( 3 − 4 2 ) ( 5 − 6 2 ) open . 3 minus 4 square root of 2 . close . open . 5 minus 6 square root of 2 . close
26. ( 3 + 5 ) 2 open , square root of 3 plus square root of 5 , close squared
27. ( 13 + 6 ) 2 open , square root of 13 plus 6 , close squared
28. ( 2 5 + 3 2 ) 2 open . 2 square root of 5 plus 3 square root of 2 . close squared

See Problem 5.

Multiply each pair of conjugates.

29. ( 5 − 11 ) ( 5 + 11 ) open , 5 minus square root of 11 , close . open , 5 plus square root of 11 , close
30. ( 4 − 2 3 ) ( 4 + 2 3 ) open . 4 minus 2 square root of 3 . close . open . 4 plus 2 square root of 3 . close
31. ( 2 6 + 8 ) ( 2 6 − 8 ) open . 2 square root of 6 plus 8 . close . open . 2 square root of 6 minus 8 . close
32. ( 3 + 5 ) ( 3 − 5 ) open , square root of 3 plus square root of 5 , close . open , square root of 3 minus square root of 5 , close

See Problem 6.

Rationalize each denominator. Simplify your answer.

33. 4 1 + 3 fraction 4 , over 1 plus square root of 3 end fraction
34. 4 3 3 − 2 fraction 4 , over 3 square root of 3 minus 2 end fraction
35. 5 + 3 2 − 3 fraction 5 plus square root of 3 , over 2 minus square root of 3 end fraction

36. 3 + 8 2 − 2 8 fraction 3 plus square root of 8 , over 2 minus 2 square root of 8 end fraction

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