LESSON 44 (B.3-B.4) Radical Equations (including Equations with Fractional Exponents)
I. Radical Equations.
To solve radical equations, you need to “undo” the radical. To undo a square root, you square; to undo a
cube root, you cube; etc. You must isolate a single radical on one side of the equation first.
You must also be aware that raising both sides of an equation to an even power (to “undo” radicals with even
root indices) can create extraneous solutions. Thus, you must check your solutions to such equations.
Solve for x .
2x 1  3
Example 1.

2x 1

2
Example 2.
3
2 x 1  9
Check:
2(5)  1  3
2 x 10

2
2
10  1  3
x
 9
2
 x
2     9  2
2
9 3
x5
yes
x  18
5
Example 3.
x
 1  2
2
 x 
3
 3  1    2 
 2 
x
1   8
2
1  1
 32
1  1
2 x  10
3
3  2x
3 4
2
5
No need to check,
it's an odd root.
3  2x
3  4
2
3  3
3  2x
1
2
 5 3  2x 
2 
  1 2
2 

5
5

5
3  2x  2
3  2x

5
 25
3  2 x  32
3
3
2 x  29
2 x 29

2
2
x
29
2
No need to check,
it's an odd root.
2x  6  x  4  1
2x  6  x  4  1
Example 5.
 x4
Check:
 x4
2(5)  6  5  4  1
2x  6  1 x  4


  1 
2 x  6   1 
2x  6
2
2
10  6  9  1

x  4 1 
2
x4
x4

2x  6  1 2 x  4  x  4
16  9  1
43 1
yes, so 5 is OK!
2 x  6  x 5  2 x  4
Check:
 x  5  x 5
2(3)  6  3  4  1
x 1  2 x  4
6  6  1  1
 x  1   2 x  4 
 x  1 x  1  4  x  4 
2
2
0  1 1
0 1  1
NO, so -3 is extraneious!
x  2 x  1  4 x 16
2
 4 x  16 4 x 16
x  2 x  15  0
( x  5)( x  3)  0
2
x  5 or x  3
II. Equations with Fractional Exponents.
When equations involve expressions having fractional exponents, you should realize that these equations are
merely radical equations “in disguise.” If you convert the equations to radical equation from, you can use the
techniques for solving radical equations.
Solve for x .
2
( x  1) 3  4
Example 6.

2
( x  1) 3

3
2
  4 2
2 3
( x  1) 3 2 
( x  1)1  23
x 1  8
1  1
x7
3
3
 4
3
3
( x  3) 2  5  13
Example 7.
No need to check,
it's an odd root.
(denominator of
the fractional power)
( x  3) 2 5  13
5
5
 ( x  3) 
( x  3)
32
2 3
2
3

 8
 8
3
3
(1  3) 2  5  13
3
4 2  5  13
3
( x  3) 2  8
3
2
Check:
2
3
2
 4
3
8
23  8
Yes, so it's OK!
( x  3)1  2 2
x3 4
x 1
ASSIGNMENT 44 (B.3-B.4): Pages A60-A62 (85, 86, 165-170, 172-175, 189, 190, 198, 108, 148, 158, 179)