Properties of Radicals

Exponents and radicalsLaws of ExponentsProperties of RadicalsScientific Notation

Roots

nth root

\sqrt[n]{a} = b \Leftrightarrow b^{n} = a

Square root of a square

\sqrt{a^{2}} = | a |

any real a

Even root of a power

\sqrt[n]{a^{n}} = | a |

n even

Odd root of a power

\sqrt[n]{a^{n}} = a

n odd

Product

\sqrt{a} \cdot \sqrt{b} = \sqrt{a b}

a ≥ 0, b ≥ 0

Product, nth roots

\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a b}

Quotient

\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}

a ≥ 0, b > 0

Quotient, nth roots

\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}

Like radicals

c \sqrt{a} \pm d \sqrt{a} = (c \pm d) \sqrt{a}

Square root

\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}

a > 0

nth root

\frac{1}{\sqrt[n]{a^{k}}} = \frac{\sqrt[n]{a^{n - k}}}{a}

0 < k < n

Conjugate, a plus a root

\frac{1}{a + \sqrt{b}} = \frac{a - \sqrt{b}}{a^{2} - b}

a² ≠ b

Conjugate, a minus a root

\frac{1}{a - \sqrt{b}} = \frac{a + \sqrt{b}}{a^{2} - b}

a² ≠ b

Conjugate, two roots

\frac{1}{\sqrt{a} + \sqrt{b}} = \frac{\sqrt{a} - \sqrt{b}}{a - b}

a ≠ b

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