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:
To solve a radical equation, we will use the Principle of Powers.
Principle â€” Principle of Powers
If a = b, then a^{n} = b^{n}
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 a^{n} = b^{n} then a = b may or may not be true.
For example, consider the following equations.

Equation A 
Equation B 
Original equation.


= 5 

= 5 
Square both sides.


= (5)^{2} 

= (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,
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,
