The properties of radicals are the calculation rules that involve expressions with radicals.

### Properties

(1) The nth root of a product of factors is equal to the product of the nth roots of each factor and vice versa :

• $$\sqrt[n]{ab}$$ = $$\sqrt[n]{a} × \sqrt[n]{b}$$, for $$a$$, $$b$$ ∈ $$\mathbb{R_{+}}$$ or ab ∈ $$\mathbb{R_{-}}$$
• Example : $$\sqrt[3]{8 × 64}$$ = $$\sqrt[3]{8}$$ × $$\sqrt[3]{64}$$ = 2 × 4 = 8

(2) The nh root of a quotient is equal to the quotient of the nth roots of the two terms of the fractional expression and vice versa :

• $$\sqrt[n]{\dfrac{a}{b}}$$ = $$\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$$, for $$a$$, $$b$$ ∈ $$\mathbb{R_{+}}$$ or ab ∈ $$\mathbb{R_{-}}$$ et $$b$$ ≠ 0
• Example : $$\sqrt[4]{\dfrac{16}{1296}}$$ = $$\dfrac{\sqrt[4]{16}}{\sqrt[4]{1296}}=\dfrac{2}{6}=\dfrac{1}{3}$$

(3) The mth root of the nth root of a number $$a$$ is equal to the root of the number $$a$$, this root having the product $$mn$$ of the indices as an index :

• $$\sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}$$
• Example : $$\sqrt[3]{\sqrt[2]{64}}$$ = $$\sqrt[6]{64}$$ = 2

(4) To raise the nth root of a number $$a$$ to the power of $$m$$, $$m$$ and $$n$$ being relatively prime, the radicand is raised to that power :

• $$\sqrt[n]{a^{m}}=\left(\sqrt[n]{a}\right)^{m}=a^{frac{m}{n}}$$
• Example : $$\sqrt[2]{3^{8}}$$ = ($$\sqrt[2]{3})^{8}$$ = $$3^{\frac{8}{2}}=3^{4}$$ = 81

(5) The value of a radical is not changed by multiplying the index of the radical and the exponent of the radicand by the same integer or by dividing them by the same divisor :

• $$\sqrt[n]{a^{m}} = \left(\sqrt[nq]{a}\right)^{mq}$$ where $$a$$ ≥ 0
• Example : $$\sqrt[4]{3^{6}}=\sqrt[8]{3^{12}}$$
• Example : $$\sqrt[4]{3^{6}}=\sqrt[2]{3^{3}}$$