Properties of 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 : \(\sqrt[4]{3^{6}}=\sqrt[2]{3^{3}}\)