10-3 Operations With Radical Expressions

Quick Review

You can use the properties of real numbers to combine radical expressions. To simplify radical expressions such as 2 5 + 3 , fraction 2 , over square root of 5 plus 3 end fraction . comma  multiply the numerator and denominator by the conjugate of the denominator, 5 − 3 . square root of 5 minus 3 , .

Example

What is the simplified form of 2 5 5 + 2 ? fraction 2 square root of 5 , over square root of 5 plus 2 end fraction . question mark

2 5 5 + 2 = 2 5 5 + 2 ⋅ 5 − 2 5 − 2 Multiply by  5 − 2 5 − 2   = 2 5 ( 5 − 2 ) ( 5 + 2 ) ( 5 − 2 ) Multiply fractions .   = 10 − 4 5 1 Simplify the numerator and denominator .   = 10 − 4 5 Simplify the fraction . table with 4 rows and 3 columns , row1 column 1 , fraction 2 square root of 5 , over square root of 5 plus 2 end fraction , column 2 equals . fraction 2 square root of 5 , over square root of 5 plus 2 end fraction . dot . fraction square root of 5 minus 2 , over square root of 5 minus 2 end fraction , column 3 cap multiplyby . fraction square root of 5 minus 2 , over square root of 5 minus 2 end fraction , row2 column 1 , , column 2 equals . fraction 2 square root of 5 open square root of 5 minus 2 close , over open square root of 5 plus 2 close open square root of 5 minus 2 close end fraction , column 3 cap multiplyfractions . . , row3 column 1 , , column 2 equals . fraction 10 minus 4 square root of 5 , over 1 end fraction , column 3 cap simplifythenumeratoranddenominator . . , row4 column 1 , , column 2 equals 10 minus 4 square root of 5 , column 3 cap simplifythefraction . . , end table

Exercises

Simplify each radical expression.

31. 5 6 − 3 6 5 square root of 6 minus 3 square root of 6
32. 2 ( 8 + 6 ) square root of 2 . open , square root of 8 plus square root of 6 , close
33. ( 3 2 − 2 5 ) ( 4 2 + 2 5 ) open . 3 square root of 2 minus 2 square root of 5 . close . open . 4 square root of 2 plus 2 square root of 5 . close
34. 3 2 − 3 fraction 3 , over square root of 2 minus 3 end fraction
35. 3 − 3 3 + 3 fraction square root of 3 minus 3 , over square root of 3 plus 3 end fraction
36. Geometry A golden rectangle is 3 in. long. The ratio of its length to its width is ( 1 + 5 ) : 2 . open , 1 plus square root of 5 , close . colon 2 . .  What is the width of the rectangle? Write your answer in simplified radical form.

10-4 Solving Radical Equations

Quick Review

You can solve some radical equations by isolating the radicals, squaring both sides of the equation, and then testing the solutions.

Some solutions may be extraneous. Some equations may have no solution.

Example

What is the solution of x + 16 = 9 x square root of x plus 16 end root , equals , square root of 9 x end root ?

x + 16 = 9 x   ( x + 16 ) 2 = ( 9 x ) 2 Square each side . x + 16 = 9 x Simplify . 16 = 8 x Subtract  x  from each side . 2 = x Divide each side by  8. table with 5 rows and 3 columns , row1 column 1 , square root of x plus 16 end root , column 2 equals , square root of 9 x end root , column 3 , row2 column 1 , open , square root of x plus 16 end root , close squared , column 2 equals . open , square root of 9 x end root , close squared , column 3 cap squareeachside . . , row3 column 1 , x plus 16 , column 2 equals 9 x , column 3 cap simplify , . , row4 column 1 , 16 , column 2 equals 8 x , column 3 cap subtract . x . fromeachside . . , row5 column 1 , 2 , column 2 equals x , column 3 cap divideeachsideby . 8. , end table

Check

2 + 16 = ? 9 ( 2 ) Substitute  2  for  x . 18 = 18 ✓   table with 2 rows and 3 columns , row1 column 1 , square root of 2 plus 16 end root , column 2 modified equals with question mark above . square root of 9 , open 2 close end root , column 3 cap substitute . 2 , for , x . , row2 column 1 , square root of 18 , column 2 equals square root of 18 check mark , column 3 , end table

The solution is 2.

Exercises

Solve each radical equation. Check your solution. If there is no solution, write no solution.

37. x − 5 = 8 square root of x minus 5 equals 8
38. 4 + y = 7 4 plus square root of y equals 7
39. w − 2 = 4 square root of w minus 2 end root , equals 4
40. f + 4 = 5 square root of f plus 4 end root , equals 5
41. 2 + d = d square root of 2 plus d end root , equals d
42. 2 r = 3 r + 1 2 square root of r equals . square root of 3 r plus 1 end root
43. n 2 = 9 − 3 n n square root of 2 equals . square root of 9 minus 3 n end root
44. 2 x = 2 − 2 x 2 x equals . square root of 2 minus 2 x end root
45. Geometry The radius r of a cylinder is given by the equation r = V π h , r equals , square root of fraction v , over pi h end fraction end root . comma  where V is the volume and h is the height. If the radius of a cylinder is 3 cm and the height is 2 cm, what is the volume of the cylinder? Round to the nearest tenth of a cubic centimeter.

Table of Contents