# Other Types of Equations

**Equations with Radical and Rational Exponents **

When solving an equation linear in form with radicals or

rational exponents , simplify the equation , isolate the most

complicated radical on one side, and raise both sides to

the power equal to the index of the radical in order to

eliminate the last. You may need to repeat this procedure

if the resulting equation still contains a radical.

Note: We solve radical equations over the real numbers .

**Caution!** If you raise both sides of an equation to
an

even power, the new equation may have more real

solutions than the original one.

**Example:** Equation x = 6 has solution set: {6}.

Raising both sides to the power 2, gives the equation

x^2 = 36 which has solution set x = ±6.

Thus, x = −6 is an extraneous solution to the original

equation and must be rejected.

**Important!** When raising to an even power, always

check each proposed solution in the original equation.

**Example: **Solve

**Example:** Solve

**Note:** If n is an even number,
is never negative .

**Example: **Solve the equation

**Example:** Solve the equation

**Equations Quadratic in Form **

**Example: **Solve by using a substitution.

**Equations Quadratic in x^2 (Biquadratic)**

**Example:**

Solve the equation in the complex number system

**Factorable Equations**

**Example: **Find all solutions of the equation:

**Example: **Solve by factoring

**Equations with Absolute Value **

Recall:

1.|x| is the distance on the number line from 0 to x.

2.|x|≥ 0

3.|x|= 0 if and only if x = 0

4.|x|= −x

5. The algebraic definition :

If a is a positive real number and u is any algebraic

expression , then

|u|= a is equivalent to u = a or u = −a .

**Example:** Solve the equations.

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