Solving Equations with Radicals and Exponents
Raising Each Side to a Power
If we start with the equation x = 3 and square both sides, we get x2
= 9. The solution
set to x2 = 9 is {-3, 3}; the solution set to the original equation is
{3}. Squaring
both sides of an equation might produce a nonequivalent equation that has more
solutions than the original equation. We call these additional solutions extraneous
solutions. However, any solution of the original must be among the solutions to the
new equation.
Caution
When you solve an equation by raising each side to a power,
you must check your answers. Raising each side to an odd power will always give
an equivalent equation; raising each side to an even power might not.
Example
Raising each side to a power to eliminate radicals
Solve the following equation:
Solution
Eliminate the square root by raising each side to the power 2:
 |
= 0 |
Original equation |
 |
= 5 |
Isolate the radical. |
 |
= 52 |
Square both sides. |
| 2x - 3 |
= 25 |
|
| 2x |
= 28 |
|
| x |
= 14 |
|
Check by evaluating x = 14 in the original equation:
The solution set is {14}.
|
