Problem 104 - Algebra and Trigonometry

Question

Simplify by reducing the index of the radical. $$ \sqrt[4]{x^{12}} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt[4]{x^{12}}\) is \(x^3\).

Step 1: Identify the index and the exponent

Look at the given expression \(\sqrt[4]{x^{12}}\). Here, the index of the radical (root) is 4, and the exponent of the variable x is 12.

Step 2: Apply the rule of radicals

You can simplify the radical by applying the rule: \(\sqrt[n]{a^m} = a^{m/n}\). Here, n=4 (index) and m=12 (exponent). So applying the rule gives you \(x^{12/4}\) (variable to the power of exponent divided by index).

Step 3: Simplify the exponent

Simplify the exponent by performing the division. This gives you \(x^3\).

Key Concepts

Index of a Radical

Understanding the index of a radical is fundamental when dealing with radical expressions. Radicals are often expressed with a small number (the index) above the radical symbol, indicating the degree of the root. For example, in the expression \(\sqrt[4]{x^{12}}\), the number 4 is the index. This tells us that we are dealing with a fourth root.
Any time you're working with radicals, remember that the index helps determine the type of root you are extracting:

• Square root when the index is 2 (often left out as in \(\sqrt{a}\)).
• Cube root when the index is 3.
• Fourth root, as in our example, when the index is 4.

Being familiar with the radical index is crucial because it helps decide how the expression can be manipulated or simplified.

Exponent Rules

Exponent rules offer a set of guidelines for managing expressions involving powers. One key rule is the rule of radicals which is particularly handy for simplifying when variables or numbers are raised to a power within a radical.
A critical exponent rule states:

• \(\sqrt[n]{a^m} = a^{m/n}\)

In our example, this rule converts \(\sqrt[4]{x^{12}}\) into something more manageable by rewriting it as a power \(x^{12/4}\).
Why is this useful? It converts a complicated radical into a simpler exponent form, making it straightforward to see the final simplified result. Other exponent rules, such as multiplying powers with the same base (\(x^a \cdot x^b = x^{a+b}\)), provide additional frameworks for breaking down complex calculations.

Simplifying Expressions

Simplifying expressions involves breaking down complex equations into their simplest form, making them easier to work with. Let's tie it together using both the index of a radical and exponent rules.
Consider the expression \(\sqrt[4]{x^{12}}\):

• First, identify the index (4) and the exponent (12).
• Use the rule of radicals: \(\sqrt[n]{a^m} = a^{m/n}\).
• Transform the expression to \(x^{12/4}\).
• Simplify the expression by calculating the exponent: \(12/4 = 3\).
• The resultant simplified expression is \(x^3\).

Simplifying expressions helps in achieving a cleaner, more interpretable form, often a necessary step in solving algebraic equations or real-world problems in mathematics.