Basic functions: Power functions
Equations with power functions
Solving an equation with power functions Finding real solutions of an equation with a power function of the form is always approached in the same way, namely ``isolate and then eliminate the -th power of by taking the -th power." To isolate the correct power, you sometimes have to convert square root notation in an equation into power notation You also have to keep an eye on whether you do not lose or introduce solutions in between.
For an equation involving a power function with a natural number unequal to as its exponent, the solution method boils down to using the -th root of the power. You usually need a calculator to compute numerical values.
Power equation with a natural number greater than 1 as exponent The graph of the function with a natural number greater than depends on whether the exponent is even or odd. You can see this in the interactive diagram below. In this diagram, also the horizontal line , for a certain number , can be manipulated. By playing with this it is easy to see how many real solutions the equation has.
even exponent (even )
The equation has no solution if .
There are two solutions if , viz., .
If , then the only solution is .
odd exponent (odd )
The equation always has one solution.
If , then .
If , then .
Power equation with a positive irreducible fraction as exponent The graph of the function with a positive irriducible fraction as exponent depends on whether is even or odd, and for odd also on whether is even or odd. You can see this in the interactive diagram below. In this diagram, also the horizontal line , for a certain number , can be manipulated. By playing with this you can quickly see how many real solutions the equation has.
even denominator in the exponent (even )
The equation has no solution if .
If , then one solution: .
If , then one solution: .
odd denominator in the exponent (odd )
If , then .
If and odd, then .
If and are even, then two solutions: and .
If and odd, then .
If and are even, then there are no solutions.
Plugging into the original equation shows that the solutions found are valid. The exponentiation in the solution method has not introduced an invalid solution.