Sometimes a variable in an equation may include also an exponent, for example [math]\Large x^4=16.[/math] [br][br]An equation [br][br]   [math]\LARGE \textcolor{blue}{x^n=a},[/math][br][br]where [i]n[/i] is a positive integer and [i]a [/i]is a real number, is called as a[color=#0000ff] power equation[/color]. If [i]n[/i] is an even natural number, then [i]a[/i] must be at least zero. For odd [i]n[/i], there is no restrictions. [br][br][table] [tr][br] [td][b][i]n[/i][/b][/td][br] [td][b][i]a[/i][/b][/td][br] [td][b][i]x[/i][/b][/td][br][/tr][br] [tr][br] [td]even [/td][br] [td][math] \Large a\geq 0[/math][br][/td][br] [td][math]\Large\pm [/math] [i]n[/i]th rooth of [i]a[/i][/td][br][/tr][br] [tr][br] [td]odd[/td][br] [td]any real number [/td][br] [td] [i]n[/i]th rooth of [i]a[/i][/td][br][/tr][br][/table][br][br][u]During this course[/u], we are concentrating on positive solutions, so [u]negative solutions can be omitted[/u].

[math]\Large\begin{eqnarray}& x^2=9 &\Rightarrow& x&=&\sqrt 9=3\\[br]&x^3=64&\Rightarrow& x&=&\sqrt[3]{64} =4[br]\end{eqnarray}[/math]

Only the square root and cube root are usually found from calculators. All the other roots can be solved with fraction power:[br][br]   [math]\huge \textcolor{blue}{\sqrt[n] a=a^\frac{1}{n}.}[/math]

[math]\Large\begin{eqnarray}[br]a^4=16&&a=\sqrt[4]{16}=16^\frac{1}{4}=2&|\text{You can also take square root twice}\\[br]x^5=1024&&a=\sqrt[5]{1024}=1024^\frac{1}{5}=4[br]\end{eqnarray}[br][/math]

[math]\Large\begin{eqnarray}[br]2\cdot x^5-60&=&4\\[br]2\cdot x^5&=&4+60\\[br]x^5&=&\frac{64}{2}=32\\[br]x&=&32^\frac{1}{5}=2 [br]\end{eqnarray}[/math]