Powers
The Principle of Powers
A radical equation is an equation in which a variable appears in a
radicand.
For example, the variable x is in the radicand of the following radical
equations:
![](./articles_imgs/855/powers13.gif)
To solve a radical equation, we will use the Principle of Powers.
Principle — Principle of Powers
If a = b, then an = bn
Here, a, b, and n are real numbers.
While the Principle of Powers is true for all real numbers, the reverse is
not always true. That is, if an = bn then a = b may or may not be true.
For example, consider the following equations.
|
Equation A |
Equation B |
Original equation.
|
![](./articles_imgs/855/powers14.gif) |
= 5 |
![](./articles_imgs/855/powers14.gif) |
= -5 |
Square both sides.
|
![](./articles_imgs/855/powers15.gif) |
= (5)2 |
![](./articles_imgs/855/powers15.gif) |
= (-5)2 |
Simplify. |
x |
= 25 |
x |
= 25 |
Note that
is the principle or positive
square root of x. This is a positive number
or zero.
For example,
![](./articles_imgs/855/powers16.gif)
The solution of Equation A, x = 25, checks since
.
However, the solution of Equation B, x = 25, does NOT check
since
Squaring both sides of Equation B introduced an extraneous solution or
false solution. Thus, when solving a radical equation we must check the
answer to verify that it satisfies the original equation.
Note:The negative square root of x is written as
- . This is a negative number.
For example,
![](./articles_imgs/855/powers19.gif)
|