Editing Elementary algebra (section)

=== Radical equations ===
{{Image frame|align=right|width=150|caption=Radical equation showing two ways to represent the same expression. The triple bar means the equation is true for all values of ''x''|content=<math>\overset{}{\underset{}{\sqrt[2]{x^3} \equiv x^{\frac 3 2} } }</math>}}
A radical equation is one that includes a radical sign, which includes [[square root]]s, <math>\sqrt{x},</math>  [[cube root]]s, <math>\sqrt[3]{x}</math>, and [[nth root|''n''th roots]], <math>\sqrt[n]{x}</math>. Recall that an ''n''th root can be rewritten in exponential format, so that <math>\sqrt[n]{x}</math> is equivalent to <math>x^{\frac{1}{n}}</math>. Combined with regular exponents (powers), then  <math>\sqrt[2]{x^3}</math> (the square root of {{mvar|x}} cubed), can be rewritten as <math>x^{\frac{3}{2}}</math>.<ref>John C. Peterson, ''Technical Mathematics With Calculus'', Publisher Cengage Learning, 2003, {{ISBN|0766861899}}, 9780766861893, 1613 pages, [https://books.google.com/books?id=PGuSDjHvircC&dq=%22+radical+equation%22&pg=PA525 page 525]</ref> So a common form of a radical equation is <math> \sqrt[n]{x^m}=a</math> (equivalent to <math> x^\frac{m}{n}=a</math>) where {{mvar|m}} and {{mvar|n}} are [[integer]]s. It has [[real number|real]] solution(s):
{| class="wikitable" style="text-align:center"
|- style="vertical-align:top"
!{{mvar|n}} is odd
!{{mvar|n}} is even<br />and <math>a \ge 0</math>
!{{mvar|n}} '''and''' {{mvar|m}} are '''even'''<br />'''and''' <math>a<0</math>
!{{mvar|n}} '''is even''', {{mvar|m}} '''is odd''', '''and''' <math>a<0</math>
|-
|<math>x = \sqrt[n]{a^m}</math>

equivalently

:<math>x = \left(\sqrt[n]a\right)^m</math>
|<math>x = \pm \sqrt[n]{a^m}</math>

equivalently

:<math>x = \pm \left(\sqrt[n]a\right)^m</math>

|<math>x=\pm \sqrt[n]{a^m}</math>
|no real solution
|}

For example, if:

:<math>(x + 5)^{2/3} = 4</math>

then

: <math>\begin{align}
x + 5 & = \pm (\sqrt{4})^3,\\
x + 5 & = \pm 8,\\
x &  = -5 \pm 8,
\end{align}</math>
and thus 
:<math>x = 3 \quad \text{or}\quad x = -13</math>