## The nth degree root

The nth degree root of a number can be found using the following formula: \( \sqrt[n]{a} = a^\frac{1}{n} \) Where "\(a\)" is the number to be rooted, and "\(n\)" is the degree of the root.

For instance, the fourth degree root of 81 can be found using the formula above:

\( \sqrt[4]{81}=81^\frac{1}{4} \)

We can simplify the expression above by using the fact that 81 is equal to 3 raised to the fourth power:

\( \sqrt[4]{81}=(3^4)^\frac{1}{4}=3^{4\cdot \frac{1}{4}}=3^1=3 \)

Therefore, the fourth degree root of 81 is 3.

It's worth noting that some nth degree roots can be irrational, meaning they cannot be expressed as a ratio of two integers. For example, the square root of \(2 (\sqrt{2}) \) is an irrational number because it cannot be written as a fraction.

**Properties of nth Roots:**

**Product Property:**The nth root of a product is equal to the product of the nth roots of the factors. That is, for any non-negative real numbers \(a\) and \(b\) and any positive integer \(n\), \( \sqrt[n]{ab}=\sqrt[n]{a}\cdot \sqrt[n]{b} \)

For example, \( \sqrt{4\cdot 9}=\sqrt{4}\cdot \sqrt{9}=2\cdot 3=6 \).**Quotient Property:**The nth root of a quotient is equal to the quotient of the nth roots of the numerator and denominator. That is, for any non-negative real numbers \(a\) and \(b\) where \(b \neq 0 \) and any positive integer \(n\), \( \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}} \)

For example, \( \sqrt{\frac{9}{4}}=\frac{\sqrt{9}}{\sqrt{4}}=\frac{3}{2} \)**Power Property:**The nth root of a power is equal to the power of the nth root. That is, for any non-negative real number \(a\) and any positive integers \(m\) and \(n\), \( \sqrt[n]{a^m}=a^\frac{m}{n}\)

For example, \( \sqrt[3]{8^2}=8^\frac{2}{3}=4\)**Radicand Property:**If \(n\) is odd, then every non-negative real number has a unique nth root. If \(n\) is even, then the nth root of a non-negative real number is defined only for non-negative radicands.

For example, \( \sqrt[3]{-8} = -2 \), but \( \sqrt{16} = 4 \) and \( \sqrt{-16} \) is not defined among the real numbers.**Exponentiation:**Exponentiation of the nth degree root is a property that tells us how to raise an nth root to a power. Specifically, for any positive integer \(n\), any non-negative real numbers \(a\) and \(b\), and any integer \(m\), \( (\sqrt[n]{a})^m = \sqrt[n]{a^m} \)

This property means that we can simplify expressions like \( (\sqrt[3]{2})^2 \) by first raising the nth root to the power, and then taking the nth root of the result: \( (\sqrt[3]{2})^2 = \sqrt[3]{2^2}=\sqrt[3]{4} \)

We can also use this property to simplify more complicated expressions involving radicals. For example, we can simplify the expression \( \sqrt[3]{2^5 \cdot 3^2} \) as follows: \( \small \sqrt[3]{2^5 \cdot 3^2}=\sqrt[3]{2^3 \cdot 2^2 \cdot 3^2}=\sqrt[3]{8\cdot 9\cdot 4}=\sqrt[3]{288} \)**The root of the root of the nth degree:**The root of the nth degree root is not a commonly used property or formula. However, one interpretation of this phrase could be the following:

For any positive integers \(m\) and \(n\) and any non-negative real number \(a\), we have: \( \sqrt[m]{ \sqrt[n]{a} }=\sqrt[mn]{a} \)

This property tells us that taking the mth root of the nth root of a non-negative real number \(a\) is equivalent to taking the mnth root of \(a\). For example, we have: \( \sqrt[2]{ \sqrt[3]{8} }=\sqrt[2\cdot 3]{2^3}=\sqrt{2} \)

**Note that this property only holds when a is non-negative**, since the nth root of a negative number is not well-defined for even values of \(n\). Additionally, while this property can be useful for simplifying certain expressions involving radicals, it is not as widely applicable as some of the other properties and formulas discussed earlier.

These properties and formulas of nth root are useful for simplifying and solving problems involving radicals.