Question

In the symbol \(\sqrt[n]{a},\) the integer \(n\) is called the _________

Short answer

index

Step-by-step solution

Identify the components of the expression

In the expression \(\sqrt[n]{a}\), \(a\) is the radicand, and \(n\) is the integer that defines the root.

Understand the terminology

The integer \(n\) that specifies the degree of the root in the expression \(\sqrt[n]{a}\) is known by a specific term in mathematics.

Recall the correct term

In mathematical terminology, the integer \(n\) is called the 'index' or 'degree' of the root.

Key concepts

Radical Expressions

Radical expressions are mathematical expressions that contain a root. The most common types of radical expressions involve square roots and cube roots.

For example, in the expression \(\sqrt{25}\), 25 is the number under the radical sign, and the result will be 5. This is because 5 squared (5 times 5) equals 25.

Other types of roots include cube roots, fourth roots, and so on. These are typically denoted with a small number above the radical sign, like this: \(\sqrt[3]{27}\).

Radical expressions allow us to work with numbers in an insightful way, especially when dealing with powers and roots.

Index of a Radical

In the world of radical expressions, the integer right above the radical sign is called the 'index'.

For example, in the expression \(\sqrt[4]{81}\), the number 4 is the index.

The index tells you what root is being taken. If there's no number specified, it's understood to be a square root, or an index of 2. So, the expression \(\sqrt{9}\) is actually \(\sqrt[2]{9}\).

Using different indexes can help simplify or solve various mathematical problems.

Degree of Root

The degree of the root refers to the index number. Essentially, it's another way of specifying which root you are working with.

For instance, in \(\sqrt[3]{8}\), the degree of the root is 3. This means you're looking for the cube root of 8, which is 2, because \(2^3 = 8\).

The degree of the root is crucial for understanding the complexity and nature of different radical expressions and for performing correct mathematical operations on them.