COLLEGE ALGEBRA
MATH 161
Section 1.4 PART 1
TEAGUE
Radical Equations
𝒎
𝒏
𝒎
𝒂 ⁄𝒏 = √𝒂
Numerator is the power and the denominator is the index.
-92 = -81
(-9)2 = 81
√(−𝟑)𝟐 = 𝟑
𝟐𝟓
𝟏⁄
𝟐
=𝟓
-(9)2 = -81
𝟏𝟎𝟎𝟎
𝟏⁄
𝟑
𝟔−𝟏 =
= 𝟏𝟎
𝟏
𝟔
25 3⁄ 125
Example 1) ( ) 2
𝑝𝑢𝑡 𝑖𝑛 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑜𝑟 𝑢𝑠𝑖𝑛𝑔 𝐴 𝑏⁄𝑐 𝑘𝑒𝑦 𝑎𝑠 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠
64
512
3
3
2
Example 2) (−16) ⁄2 = 𝑚𝑒𝑎𝑛𝑠 √−16
𝑛𝑜 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑜𝑓 𝑎 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑛𝑢𝑚𝑏𝑒𝑟
2
3
2
Example 3) (−512) ⁄3 = 𝑚𝑒𝑎𝑛𝑠 √−512 =-64 because you can take cube root
of a negative number
Example 4) 64−
5⁄
3
*calculator work compute negative
64
Then negative exponent makes it
Example 5) 𝑥 = 12√𝑥
Example 6) √2𝑥 − 4 = −6
Square both sides 𝑥 2 = 144𝑥
𝑥 2 − 144𝑥 = 0
x = 0, 144
5⁄
3 =1024
1
1024
x(x-144) = 0
NO SOLUTION BECAUSE SQUARE ROOT ≠ NEGATIVE
Example 7) 2+ √4𝑥 − 3 = 𝑥 move 2 to the right and
square both si des *use FOIL on the right
4𝑥 − 3 = 𝑥 2 − 4𝑥 + 4  0=𝑥 2 − 8𝑥 + 7 (x-7)(x-1) x = 1,7
CHECK BOTH ANSWERS WITH RADICAL PROBLEMS
2+ √4(1) − 3 = (1)
2+ √4(7) − 3 = (7)
4 ≠ 1 NO
2+5 = 7 yes
{7}
square both sides 𝑥 2 = 4(2𝑥 − 4)
𝑥 2 = 8𝑥 − 16
𝑥 2 − 8𝑥 + 16 = 0
(𝑥 − 4)(𝑥 − 4) = 0
x=4
Example 8) 𝑥 = 2√2𝑥 − 4
Example 9) √𝑥 2 − 𝑥 − 6 = 𝑥 + 4
Square both sides *use FOIL on the right
𝑥 2 − 𝑥 − 6 = 𝑥 2 + 8𝑥 + 16
−9𝑥 = 22
𝑥=−
Example 10) 𝑥
𝑥
3⁄
2
1⁄
2
− 40𝑥
=0
1⁄
2
=0
(𝑥
1⁄ 2
2)
22
9
Use u substitution u3-40u=0
u = x1/2
u(u2-40)=0
u = 0 u2=40
= 40
x = 0 x = 40  (0,40) always 0, the number in the problem