Equations with power functions

Functions: Power functions

Equations with power functions

In quadratic equations, we have seen how to solve an equation $x^2=c$ . With the same procedure, we will use higher degree roots to solve an equation $x^n=c$ .

The solutions to the equation $x^\orange{n}=\blue{c}$ are dependent on the values of $\orange n$ and $\blue c$ .

$\blue{c} \gt 0$

$\blue{c}=0$

$\blue{c} \lt 0$

$\orange{n}$ is even

Two solutions:

$x=-\sqrt[\orange{n}]{\blue{c}} \lor x=\sqrt[\orange{n}]{\blue{c}}$

One solution:

$x=0$

No solutions

$\orange{n}$ is odd

One solution:

$x=\sqrt[\orange{n}]{\blue{c}}$

One solution:

$x=0$

One solution:

$x=\sqrt[\orange{n}]{\blue{c}}$

Fractional powersPlease remember that we have seen the following rule with Fractional powers:

$\sqrt[\orange{n}]{x}=x^{\frac{1}{\orange{n}}}$

Therefore, we can also write the solutions with a fractional power. The solutions then look like this:

$\blue{c} \gt 0$

$\blue{c}=0$

$\blue{c} \lt 0$

$\orange{n}$ is even

Two solutions:

$x=-\blue{c}^{\frac{1}{\orange{n}}} \lor x=\blue{c}^{\frac{1}{\orange{n}}}$

One solution:

$x=0$

No solutions

$\orange{n}$ is odd

One solution:

$x=\blue{c}^{\frac{1}{\orange{n}}}$

One solution:

$x=0$

One solution:

$x=\blue{c}^{\frac{1}{\orange{n}}}$

In the examples, we see that you can reduce many equations to the form $x^\orange{n}=\blue{c}$ and then solve them.

Solve $4\, x^{4}+6=9$ for $x$ .

Give your answer in the form $x=x_1$ or $x=x_1 \lor x=x_2$ for suitable numbers $x_1$ and $x_2$ .

$x=\frac{1}{2}\sqrt[4]{12} \lor x=-\frac{1}{2}\sqrt[4]{12}$

$\begin{array}{rcl}4\, x^{4}+6&=& 9 \\ &&\phantom{xxx}\blue{\text{the equation we need to solve}} \\ 4\, x^{4}&=&3 \\ &&\phantom{xxx}\blue{\text{both sides minus }6} \\ x^{4} &=& {{3}\over{4}} \\ &&\phantom{xxx}\blue{\text{both sides divided by }4} \\ x=\sqrt[4]{{{3}\over{4}}} &\lor& x=-\sqrt[4]{{{3}\over{4}}} \\ &&\phantom{xxx}\blue{\text{both sides taken the }4 \text{-th root}}\\ x=\frac{1}{2}\sqrt[4]{12} &\lor& x=-\frac{1}{2}\sqrt[4]{12}\\ &&\phantom{xxx}\blue{\text{simplified}} \end{array}$

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