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Solving Linear Systems of Equations by Elimination
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Factoring a Sum or Difference of Two Cubes
Solving Nonlinear Equations by Substitution
Solving Systems of Linear Inequalities
Arithmetics with Decimals
Finding the Equation of an Inverse Function
Plotting Points in the Coordinate Plane
The Product of the Roots of a Quadratic
Solving Quadratic Equations by Completing the Square
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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 — 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.


The negative square root of x is written as -. This is a negative number.

For example,

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