Solving Radical Equations

SOLVING RADICAL EQUATIONS

The Principle of Powers

A radical equation has variables in one or more radicands—for example,

$\sqrt[3]{2x}$ + 1 = 5, $\sqrt{x}$ + $\sqrt{4x - 2}$ = 7.

To solve such an equation, we need a new equation-solving principle. Suppose

that an equation a = b is true. If we square both sides, we get another

true equation: a² = b². This can be generalized.

THE PRINCIPLE OF POWERS

For any natural number n, if an equation a = b is true, then aⁿ = bⁿ is true.

However, if an equation aⁿ = bⁿ is true, it may not be true that a = b, if n

is even. For example, 3² = (-3)² is true, but 3 = -3 is not true. Thus we must

make a check when we solve an equation using the principle of powers.