How to Solve Equations With Two Rational Exponents

Equations with two rational exponents can be solved using standard techniques.

Rational exponents, also called fractional exponents, are fractional numbers that a specific number (called the base) is raised to the power of. For example, in 3^(1/2), 3 is the base, while 1/2 is the rational exponent. The denominator of the exponent is the called the root, while the numerator of the exponent is called the exponent. Rational exponents frequently appear in equations, sometimes more than once within a single equation. Solving an equation with two rational exponents can be accomplished in a few short steps.

Instructions

1. Isolate one of the terms which contains a rational exponent. For example, if the equation you are given is

   4 = (x + 16)^(1/2) - x^(2/2)

   move one of the terms with the rational exponent (also called a radical) to the other side of the equal sign. This gives

   x + 4 = (x + 16)^(1/2)

   (Note that the expression x^(2/2) was simplified to just x in the last step).

2. Get rid of the radicals by squaring both sides of the equation. Doing this for the example above gives

   (x + 4)^2 = x + 16.

3. Simplify the expression found after getting rid of the radical. Simplifying the expression

   (x + 4)^2 = x + 16

   Gives

   x^2 + 8x + 16 = x + 16

   and further simplifying gives

   x^2 + 7x = 0.

   Call this expression A.

4. Solve expression A to find solutions of the equation. The expression

   x^2 + 7x = 0

   can be simplified to

   x (x+7) = 0.

   Therefore, the two solutions that make the original equation true are x = 0 and x = -7.

Tips & Warnings

• If the two rational exponents are both noninteger numbers (for example, 3/2, 5/3), the equation may require more advanced techniques to solve, and it may only be possible to solve the equation with the aid of a computer.

References

• West Texas A&M University: Radical Equations andEquations Involving Rational Exponents
• The Math Page: Rational Exponents