Solving an Exponential Equation: Relating the Bases

Example:  Solve the exponential equation.

Solution:

 The exponential equation Try to write both sides of the equation with the same base.  Try 4 since there is a base of 4 on the left Using a property of negative exponents move the base to the numerator Now that that the bases are the same the exponents must be equal Solve for x

The solution the the exponential equation is 4.

Rational Equation (no solution)

Example: Solve the rational equation.

Solution:

Since we are solving a rational equation we need to first find the restrictions (the values of x that cause the expression to be undefined).

To find the restrictions create an equation by setting each denominator equal to zero and solving.

Having x=3 causes a zero in the denominator and the overall expression undefined.  That makes 3 a restricted value .

With the restriction in mind we will solve the equation.

 The original equation Multiply each side of the equation by the least common multiple of the denominators.  For this equation the least common multiple is Distribute the least common multiple to each term. Simplify by canceling the common factors.  This should clear any denominators. Use the distributive property to simplify. Simplify each side of the equation by combining like terms. Solve for x by getting x by itself on one side.  Start by subtracting 1 on both sides. Solve for x by getting x by itself on one side.  Next divide both sides by 2. Compare your solution to the restricted value. Since the solution is the same as the restricted value we must exclude it as a solution.  Since all of the solutions have been excluded, there is no solution to the rational equation.

Video Example:

Higher Order Equation that reduces to a linear equation

Example: Solve the equation.

Solution:

 The original equation Simplify both sides of the equation.  On the left hand side, rewrite the exponent.  On the right hand side, begin to simplify the multiplication. Simplify both sides of the equation.  On the left hand side, begin multiplying.  On the right hand side, combine like terms. Simplify both sides of the equation.  On the left hand side, combine like terms.  On the right hand side use the distributive property. Simplify both sides of the equation.  On the left hand side, continue multiplying.  The right hand side is in simplest form. Simplify both sides of the equation.  On the left hand side, combine like terms.  The right hand side is in simplest form. Now that each side is in simplest form we want the terms with x on one side and the constant terms on the the other side.  Subtract from each side.  It cancels from each side. Subtract from each side.  It cancels from each side. Subtract from each side and simplify. Subtract from each side and simplify. Get x by it self by dividing by 19 on both sides and simplify.