Cube Root, Square Root, and Nth Root in LaTeX

Roots in LaTeX

Writing roots in LaTeX is straightforward once you know the syntax.

Square Root

The basic square root:

$\sqrt{x}$

With expressions inside:

$\sqrt{a^2 + b^2}$

$\sqrt{\frac{a}{b}}$

Cube Root

For cube roots, use the optional argument:

$\sqrt[3]{x}$

The 3 goes in square brackets before the radicand.

Examples:

$\sqrt[3]{8} = 2$

$\sqrt[3]{a^3 + b^3}$

$\sqrt[3]{\frac{27}{8}}$

Nth Root

The same pattern works for any root:

$\sqrt[n]{x}$

Common examples:

Fourth root: $\sqrt[4]{16} = 2$

Fifth root: $\sqrt[5]{32}$

General: $\sqrt[n]{a^n} = a$

Nested Roots

Roots can be nested:

$\sqrt{\sqrt{x}}$

$\sqrt[3]{\sqrt{x}}$

$\sqrt{1 + \sqrt{1 + \sqrt{1 + x}}}$

Roots in Fractions

$\frac{1}{\sqrt{2}}$

$\frac{\sqrt{3}}{2}$

Rationalizing (Root in Denominator)

Often you'll rationalize:

$\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Roots with Exponents

Combine roots and powers:

$\sqrt{x^2} = |x|$

$\left(\sqrt{x}\right)^2 = x$

$\sqrt[3]{x^6} = x^2$

Fractional Exponents (Alternative)

Roots can be written as fractional exponents:

$\sqrt{x} = x^{1/2} = x^{\frac{1}{2}}$

$\sqrt[3]{x} = x^{1/3} = x^{\frac{1}{3}}$

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

Common Formulas with Roots

Quadratic Formula

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

Distance Formula

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

Standard Deviation

$\sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{N}}$

Pythagorean Theorem

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

Styling the Radical

For a cleaner look with tall content, add spacing:

$\sqrt{\,\frac{a}{b}\,}$

The \, adds thin spaces inside.

Complex Numbers

Cube roots of unity:

$\omega = e^{2\pi i/3} = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$

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