LaTeX Square Root - Complete \sqrt Command Guide

% Simple square root
$\sqrt{x}$

% Square root of a number
$\sqrt{25} = 5$

% Square root with variables
$\sqrt{a^2 + b^2}$

% In display math mode
\[ \sqrt{x + y} \]

% Mathematical constant
$\sqrt{2} \approx 1.414$

% Pythagorean theorem
$c = \sqrt{a^2 + b^2}$

% Distance formula
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

% Quadratic formula component
$\sqrt{b^2 - 4ac}$

% General syntax
\sqrt[n]{x}  % nth root of x

% Cube root (n=3)
$\sqrt[3]{8} = 2$

% Fourth root (n=4)
$\sqrt[4]{16} = 2$

% Fifth root
$\sqrt[5]{32} = 2$

% Higher roots
$\sqrt[10]{1024}$

% Double nested square root
$\sqrt{\sqrt{x}}$

% Square root of a sum with a square root
$\sqrt{a + \sqrt{b}}$

% Triple nested
$\sqrt{1 + \sqrt{2 + \sqrt{3}}}$

% Famous mathematical constant
$\phi = \frac{1 + \sqrt{5}}{2}$ % Golden ratio

% Nested with fractions
$\sqrt{x + \frac{1}{\sqrt{y}}}$

% Multiple nested radicals
$\sqrt{a + \sqrt{b + \sqrt{c}}}$

% Ramanujan's nested radical
$\sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + 4\sqrt{1 + \cdots}}}} = 3$

% Nested with nth roots
$\sqrt{x + \sqrt[3]{y + \sqrt[4]{z}}}$

% Square root in numerator
$\frac{\sqrt{x}}{2}$

% Multiple terms
$\frac{\sqrt{a^2 + b^2}}{c}$

% Quadratic formula
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

% Square root in denominator
$\frac{1}{\sqrt{x}}$

% Rationalized form
$\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

% Complex expression
$\frac{a}{\sqrt{b^2 + c^2}}$

% Inline style (smaller, compressed)
The value is $\frac{\sqrt{x}}{2}$ which is compact.

% Display style (larger, more readable)
\[
    x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

% Force display style in inline math
The formula $\displaystyle\frac{\sqrt{x}}{2}$ is larger.

% Normal spacing (automatic)
$a + \sqrt{b}$

% Add thin space if needed
$a + \,\sqrt{b}$

% Negative thin space (rarely needed)
$a\! + \sqrt{b}$

% With parentheses
$(\sqrt{a} + \sqrt{b})^2$

% LaTeX automatically extends the radical
$\sqrt{a + b + c + d + e + f}$

% Multiple lines under square root (not recommended)
% Better to simplify or break the equation

% With equation breaking (advanced)
\begin{align}
    x &= \sqrt{(a + b + c)} \\
      &\quad \times \sqrt{(d + e + f)}
\end{align}

% Normal nth root
$\sqrt[n]{x}$

% Bold index (rarely needed)
$\sqrt[\mathbf{n}]{x}$

% Italic index (default in most cases)
$\sqrt[n]{x}$

% With subscript in index
$\sqrt[n_i]{x}$

% Standard form
\[ c = \sqrt{a^2 + b^2} \]

% With specific values
The hypotenuse is $c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

% 3D distance
\[ d = \sqrt{x^2 + y^2 + z^2} \]

% Complete quadratic formula
\begin{equation}
    x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\end{equation}

% Discriminant
The discriminant is $\Delta = b^2 - 4ac$.
\begin{itemize}
    \item If $\Delta > 0$: two real solutions $x = \frac{-b \pm \sqrt{\Delta}}{2a}$
    \item If $\Delta = 0$: one solution $x = \frac{-b}{2a}$
    \item If $\Delta < 0$: complex solutions
\end{itemize}

% Population standard deviation
\[ \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} \]

% Sample standard deviation
\[ s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2} \]

% Root mean square
\[ \text{RMS} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} x_i^2} \]

% Absolute value of complex number
\[ |z| = \sqrt{a^2 + b^2} \quad \text{where } z = a + bi \]

% Magnitude of vector
\[ \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2} \]

% Euclidean norm
\[ \|x\|_2 = \sqrt{\sum_{i=1}^{n} x_i^2} \]

% Square root of 2 (Pythagoras' constant)
$\sqrt{2} \approx 1.414213562$

% Golden ratio
$\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$

% Euler's identity relationship
$e^{i\pi/4} = \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}$

% Plastic constant
$\rho = \sqrt[3]{\frac{1}{2} + \frac{1}{6}\sqrt{\frac{23}{3}}} + \sqrt[3]{\frac{1}{2} - \frac{1}{6}\sqrt{\frac{23}{3}}}$

% WRONG - only takes square root of x
$\sqrt x + y$  % Produces: √x + y

% CORRECT - square root of entire expression
$\sqrt{x + y}$  % Produces: √(x + y)

% WRONG - causes error
The square root of 25 is \sqrt{25}.

% CORRECT - use math mode
The square root of 25 is $\sqrt{25} = 5$.

% WRONG - wrong bracket type
$\sqrt{3}{8}$  % This doesn't work

% CORRECT - use square brackets for index
$\sqrt[3]{8}$  % Cube root of 8

% TOO COMPLEX - hard to read
$\sqrt{\sqrt{\sqrt{\sqrt{x}}}}$

% BETTER - use fractional exponents
$x^{1/16}$ or $x^{\frac{1}{16}}$

% OR - introduce intermediate variables
Let $y = \sqrt{x}$, then $z = \sqrt{y}$, so $w = \sqrt{z}$.

% Variable as root index
$\sqrt[n]{x^n} = x$

% Expression as root index
$\sqrt[m+n]{x}$

% Fraction as root index (unusual but valid)
$\sqrt[\frac{1}{2}]{x} = x^2$

\usepackage{amsmath}

% Multi-line square root (rare)
\begin{equation}
\begin{split}
x = \sqrt{
    &a^2 + b^2 + c^2 \\
    &+ d^2 + e^2 + f^2
}
\end{split}
\end{equation}

% Square root in aligned equations
\begin{align}
    a &= \sqrt{b} \\
    c &= \sqrt{d + e} \\
    f &= \sqrt{g + h + i}
\end{align}

% Use \vphantom for consistent heights
$\sqrt{x} + \sqrt{y}$  % May have different heights

% Force same height
$\sqrt{x\vphantom{y}} + \sqrt{y}$  % Same height

% Useful in aligned equations
\begin{align}
    \sqrt{a\vphantom{b^2}} &= c \\
    \sqrt{b^2} &= d
\end{align}