A useful method for solving algebraic equations that contain negative exponents is to factor out a negative greatest common factor, or GCF. For example, consider the equation 3x –3 – 5x –2 = 0.
This equation has a solution that you can find without switching to fractions right away. In general, equations that have no constant terms — all the terms have variables with exponents on them — work best with this technique.
Here are the steps:
Factor out the greatest common factor (GCF).
In this case, the GCF is x –3:
3x–3 – 5x–2 = x–3(3 – 5x) = 0
Did you think the exponent of the greatest common factor was –2? Remember, –3 is smaller than –2. When you factor out a greatest common factor, you choose the smallest exponent out of all the choices and then divide each term by that common factor.
The tricky part of the factoring is dividing 3x –3 and 5x –2 by x –3. The rules of exponents say that when you divide two numbers with the same base, you subtract the exponents, so you have
Set each term in the factored form equal to zero to solve for x.
but that can never be true. The numerator has to be 0 to have a fraction be equal to 0.
Check your answer.
The only one to consider in this example is
It works!
