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 :

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

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

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

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

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

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

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