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# 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 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. = 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, 