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How to Solve Exponential Equations With Fractions

Exponential equations often require nonstandard algebra manipulation to simplify the equation. Removing variables to an exponent is often accomplished by combining or canceling the bases of the exponents. The end result is an equation that is less complex to evaluate. In some situations, however, such as limit analysis and infinite series, it is not always feasible to perform this action because there are multiple variables to account for, and often the expressions become more complex. There are, however, exponent rules to simplify exponentials that do not involve factoring and cancellation, as taught in entry-level algebra.

Instructions

    • 1

      Simplify the expression as much as possible using standard algebraic rules. For example, ensure that all functions are, if possible, expressed as multiplicative factors and that any possible trigonometric substitutions already have been carried out. Problems requiring exponential simplification are often rational expressions that provide opportunities for term cancellation.

    • 2

      Place the entire expression in parentheses.

    • 3

      Raise the entire expression in the parentheses by the inverse power of the exponents. For example,

      (e^x - e^-x) = 2.35, raised to the inverse power, (e^x - e^-x)^ 1/x = 2.35^1/x.

    • 4

      Distribute the inverse exponent and simplify. Continuing from above, (e^x - e^-x)^ 1/x = 2.35^1/x, simplifies into (e^x/x - e^-x/x) = 2.35^1/x. Further, e^1 - e^-1 = 2.35^1/x. Factoring out e^1, e^1(1 - 1/e^2) = 2.35^1/x. This takes the variable away from the exponential functions and places it as an "xth root" of a constant number, 2.35. This also provides an opportunity to differentiate or integrate the "e" terms easily as they are now just constant values, about 2.7.

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