Radicals represent expressions involving roots, such as square or cube roots, and their properties are crucial in solving equations. Given the expression \( \sqrt[\large{n}]{a} \), with \(a \geq 0\), nth root extraction translates into finding a real number \( b \) such that its nth power equals \( a \), or \( b^{\large{n}} = a\) and \(b \geq 0\). In formal terms:

\[\sqrt[\large{n}]{a} = b \quad \text{with} \quad b\geq 0 \quad \text{and} \quad b^n = a\]

The term \(a\) beneath the radical sign is the radicand, while the term \(n\) is the index of the root. For instance, if we consider the expression \(2^3 = 8\), then the cube root of \(8\) can be derived as \(\sqrt[\large{3}]{8} = 2\), because the process of extracting the cube root of \(8\) will eventually lead us back to the original value of \(2\).

This means that when we take the root of a number, we are essentially performing the opposite operation of raising that number to a power. In other words, root extraction and exponentiation are inverse operations of each other.


Properties

The following are the essential properties of roots that can aid in their manipulation and simplification in expressions. Ensuring that roots are handled appropriately can help avoid errors and ensure mathematical accuracy.

  • Taking the root of a power is equivalent to raising the number to the power divided by the root index: \[\sqrt[n]{a^m} = a^{\frac{m}{n}}\]

  • When multiplying two numbers under the same root, the result is equivalent to the product of their respective roots: \[\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}\]

  • When dividing two numbers under the same root, the result is equivalent to the root of their quotient: \[\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[\large{n}]{\frac{a}{b}}\]

  • Raising a number under a root to a power is equivalent to raising the number to the same power and then taking root: \[(\sqrt[n]{a})^m = \sqrt[\large{n}]{a^m}\] When we multiply or divide the index of a root and the exponent of the radicand by the same number \(k \geq 0 \), the value of a radical doesn’t change. \[(\sqrt[\large{n}]{a})^m = \sqrt[\large{nk}]{a^{mk}}\]

  • Taking the root of a root is equivalent to taking the root of their product: \[\left(\sqrt[m]{\sqrt[n]{a}}\right) = \sqrt[\large{mn}]{a}\]


Powers with rational exponents

In general, every number \(a \in \mathbb{R}^+\) has an nth power and an nth root. Each nth root can be rewritten as a power of \(a\) raised to the reciprocal of \(n\). Formally, we have:

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

The previous expression denotes that radicals can be treated as powers, and the same properties of powers apply to radicals expressed in the form \(a^{1}{n}\) .

Example

Simplify the following expression and write the result in radical form:

\[ \frac{ \sqrt{a}}{\sqrt[3]{a} } \]


By utilizing the properties of radicals, we can derive:

\[ \sqrt{a} = a^{\large{\frac{1}{2}}} \] \[ \sqrt[3]{a} = a^{\large{\frac{1}{3}}} \]

For the properties of exponents, we have:

\[ \frac{a^m}{a^n }= a^{m – n} \]

The expression becomes: \[ \frac{a^\frac{1}{2}}{a^\frac{1}{3}} = a^\frac{1}{2 – \frac{1}{3}} = a^{\large{\frac{1}{6}}} \]

The initial expression becomes:

\[ \frac{ \sqrt{a}}{\sqrt[3]{a} } = a^{\Large{\frac{1}{6}}} \]