What Are Power Equations?

Power equations are equations where the variable is raised to a power $n$ , for example $x^{n}$ . You solve power equations by tidying the expression as in the previous posts so that the exponential expression $x^{n}$ , is on its own and not multiplied or divided by anything. At that point you take the $n$ th root on both sides, where $n$ is the same as the exponent of $x$ . It is important to remember that if $n$ is an odd number ( $x^{3}$ , $x^{5}$ , $x^{7}$ , $\dots$ ), there will be either a negative or a positive answer. If $n$ is an even number ( $x^{2}$ , $x^{4}$ , $x^{6}$ , $\dots$ ), there is both a positive and a negative solution.

You will have a negative and a positive answer as when you multiply a negative number with itself an even number of times the answer is positive. If you multiply a negative number with itself an odd number of times the answer is negative.

Rule

Solving power equations

Even number exponential:

\begin{array}{l} x^{even number} & = number \\ x & = \pm \sqrt[even]{number} \end{array}

Odd number exponential:

\begin{array}{l} x^{odd number} & = number \\ x & = \sqrt[odd number]{number} \end{array}

Example 1

\begin{array}{l} x^{3} & = 27 \\ \sqrt[3]{x^{3}} & = \sqrt[3]{27} \\ x & = 3 \end{array}

Example 2

\begin{array}{l} 4 x^{2} & = 64 \\ 4 x^{2} & = 64 & | \div 4 \\ x^{2} & = 16 \\ \sqrt{x^{2}} & = \pm \sqrt{16} \\ x & = \pm 4 \end{array}

Example 3

\begin{array}{l} 5 x^{6} - 100 & = 220 \\ 5 x^{6} & = 320 & | \div 5 \\ x^{6} & = 64 \\ x & = \pm \sqrt[6]{64} \\ x & = \pm 2 \end{array}

Example 4

\begin{array}{l} 3 x^{5} & = 96 \\ 3 x^{5} & = 96 & | \div 3 \\ x^{5} & = 32 \\ x & = \sqrt[5]{32} \\ x & = 2 \end{array}

Example 5

\begin{array}{l} x^{5} & = - 32 \\ \sqrt[5]{x^{5}} & = \sqrt[5]{- 32} \\ x & = - 2 \end{array}

Example 6

\begin{array}{l} 2 x^{4} & = 32 \\ 2 x^{4} & = 32 & | \div 2 \\ x^{4} & = 16 \\ \sqrt[4]{x^{4}} & = \pm \sqrt[4]{16} \\ x & = \pm 2 \end{array}

Note! When you have an expression where the percentage or growth factor is unknown you often use $n$ th roots to find the answer.

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