Problem 66 - Algebra 2 and Trigonometry

Question

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ \left(x^{13}\right)^{\frac{1}{7}} $$

Step-by-Step Solution

Answer

The simplified radical expression is \( x \cdot \sqrt[7]{x^6} \).

Step 1: Understand the Problem

We are given the expression \( (x^{13})^{\frac{1}{7}} \) and asked to write this as a radical expression in its simplest form. This requires transforming the power into a radical.

Step 2: Apply Power-to-Radical Conversion

The expression \( (a^m)^{\frac{1}{n}} \) can be written as \( \sqrt[n]{a^m} \). For our expression, \( a = x \), \( m = 13 \), and \( n = 7 \), so it becomes \( \sqrt[7]{x^{13}} \).

Step 3: Simplify Using Radical Properties

The expression \( \sqrt[7]{x^{13}} \) can be simplified by noting that \( x^{13} = x^{7} \cdot x^{6} \). Therefore, \( \sqrt[7]{x^{13}} \) can be rewritten as \( (\sqrt[7]{x^{7}}) \cdot (\sqrt[7]{x^{6}}) \). Since \( \sqrt[7]{x^{7}} = x \), we end up with \( x \cdot \sqrt[7]{x^{6}} \).

Key Concepts

Power-to-Radical Conversion

When dealing with expressions that feature fractional exponents, the concept of converting powers to radicals is fundamental. A fractional exponent like \( (a^m)^{\frac{1}{n}} \) can be transformed into a radical expression using the rule \( \sqrt[n]{a^m} \). Here, \( n \) represents the index of the radical or the root, and \( m \) is the power of the base \( a \).

This technique is vital for simplifying and understanding expressions in a more intuitive form. For example, given \( (x^{13})^{\frac{1}{7}} \), we recognize it as having a fractional exponent of \( \frac{13}{7} \). Applying the conversion, we rewrite this as \( \sqrt[7]{x^{13}} \), where the seventh root is indicated by the denominator. This step is crucial because it sets the groundwork for further simplification into a clearer radical form.

Simplifying Radicals

Once a power-to-radical conversion has been made, simplifying the radical expression is the next step. Simplification involves reducing the radical expression to its simplest form by examining its factors.

Consider the example \( \sqrt[7]{x^{13}} \). To simplify, we identify that \( x^{13} \) can be broken down into \( x^{7} \cdot x^{6} \). This decomposition allows us to rewrite the expression as \( (\sqrt[7]{x^{7}}) \cdot (\sqrt[7]{x^{6}}) \).

Since \( \sqrt[7]{x^{7}} = x \) (because taking the seventh root of \( x^{7} \) cancels the exponent to one), the expression simplifies further to \( x \cdot \sqrt[7]{x^{6}} \). This process is about finding and extracting perfect roots where possible, leaving behind a cleaner and more manageable expression.

Exponent Properties

Understanding exponent properties is critical for working with radical and power expressions efficiently. The main properties used in this context are the product of powers and the power of a power rules.

The product of powers rule tells us that when multiplying like bases, we add their exponents: \( a^m \times a^n = a^{m+n} \). This helps in breaking down expressions like \( x^{13} = x^7 \times x^6 \).

The power of a power rule states that \((a^m)^n = a^{m\times n}\). This principle is useful in converting powers with fractional exponents to radicals by facilitating the rewriting of the base and exponent configuration. Such rules simplify the process of dealing with complex expressions and are the backbone for expressing powers in different forms, as seen in rewriting \( (x^{13})^{\frac{1}{7}} \) as a radical. Understanding these properties makes it easier to navigate transitions between exponential and radical forms of expressions.