# 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

**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

ex pression , then

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

**Example:** Solve the equations.

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