Solving Exponential Power Functions for X When the Exponent is a Fraction
Description/Explanation/Highlights
Video Description
This video describes how to solve a power function for a missing variable in the base when the exponent is a fraction.
Steps and Key Points to Remember
To solve power functions for a missing variable in the base, follow these steps:
- Start by doing the Algebra necessary to isolate the variable on one side of the equation i.e. In the example: \(x^{\frac{1}{2}}-4=5\) get rid of -4 that is not part of the base or power by adding 4 to both sides. This leaves: \(x^{\frac{1}{2}}=9\) and the base and power are isolated.
- Raise both sides to a reciprocal power. i.e. the reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\) or 2. \({(x^{\frac{1}{2}})}^{\frac{2}{1}}=9^{\frac{2}{1}}\)
- When you raise a power to another power by the rules of exponents, you multiply, therefore the exponents cancel on the side with the varible leaving just the variable. Simplify the other side. \(x=9^2\) therefore \(x=81\)
- If the exponent on the opposite side of the equation from the variable is a fraction, rewrite as a root and power to solve (see lesson on fractional exponents/roots and powers) or use a calculator to get the final answer. If using a calculator, be sure to put the exponent in parenthesis when entering. i.e. enter as 9^(1/2)
Here are some key points to keep in mind when solving power functions with fractional exponents:
- Isolate the base and power by getting rid of any terms added or subtracted first. Next get rid of numbers or signs in front of the variable or numbers under it.
- Be sure to raise both sides to the reciprocal power.
- If the result ends with a fractional exponent, convert to a root and power to solve or use a calculator and round the final result if necessary.
Video Highlights
- 00:00 Introduction to solving power functions
- 00:09 \(x^{\frac{1}{2}}-4=5 \) simple one-step example of isolating the variable and solving
- 01:34 \(4x^{\frac{2}{3}}+3=19 \) example with a number in front of x
- 04:10 Summary of the steps to solve a power function with a fractional exponent