Problem 46 - College Algebra with Modeling and Visualization

Question

Write the expression in radical notation. $$ \left(\frac{x}{y}\right)^{-2 / 7} $$

Step-by-Step Solution

Answer

\( \sqrt[7]{\frac{y^2}{x^2}} \)

Step 1: Understand the Expression

We are given the expression \( \left(\frac{x}{y}\right)^{-2/7} \). This is an example of a power with a negative fractional exponent, which we'll convert into radical notation.

Step 2: Rewrite the Negative Exponent

First, we rewrite the expression with a positive exponent by taking the reciprocal: \( \left(\frac{x}{y}\right)^{-2/7} = \left(\frac{y}{x}\right)^{2/7} \).

Step 3: Convert to Radical Notation

Now, convert the expression into radical form. Recall that fractional exponents can be expressed as roots: \( a^{m/n} = \sqrt[n]{a^m} \). Therefore, \( \left(\frac{y}{x}\right)^{2/7} \) becomes \( \sqrt[7]{\left(\frac{y}{x}\right)^2} \).

Step 4: Express the Final Result

Finally, write the expression clearly in radical notation: \( \sqrt[7]{\frac{y^2}{x^2}} \).

Key Concepts

Negative Fractional Exponents

Negative fractional exponents can seem intimidating at first glance, but they actually simplify processes involving roots and reciprocals greatly. A negative fractional exponent such as \( \left(\frac{x}{y}\right)^{-2/7} \) combines two key ideas:

• Negative Exponent: The negative sign indicates that the expression is a reciprocal.
• Fractional Exponent: The fraction \(-2/7\) suggests that we are dealing with a root, specifically a 7th root.

To handle such exponents, separate and tackle these ideas individually. The negative exponent suggests swapping the numerator and denominator, transforming the expression into a positive exponent. From there, focus on the fractional exponent to convert it to a root.

Reciprocal

In mathematics, a reciprocal refers to the inverse of a number or expression. You obtain it by flipping the numerator and the denominator of a fraction. For example, the reciprocal of \( \frac{x}{y} \) is \( \frac{y}{x} \).
When dealing with negative exponents, the reciprocal is a critical component. A negative exponent, such as in \( \left(\frac{x}{y}\right)^{-2/7} \), requires us to find the reciprocal first to convert the exponent into a positive one. Therefore, \( \left(\frac{x}{y}\right)^{-2/7} \) becomes \( \left(\frac{y}{x}\right)^{2/7} \).
Understanding and using reciprocals helps simplify the overall expression further.

Roots

Roots are fundamental operations in mathematics, often indicated using radical notation or fractional exponents. Given an expression like \( a^{m/n} \), the denominator \( n \) represents the root. Here, it suggests taking the nth root of \( a \).
In our case, for \( \left(\frac{y}{x}\right)^{2/7} \), the 7 in the denominator of the fractional exponent signifies a 7th root. Essentially, it indicates that the expression within the radical should be broken into something that, when multiplied by itself seven times, equals the expression’s base.
Finding roots helps simplify expressions, making them more manageable and easier to work with.

Radial Form

Radical notation refers to expressing an expression using the radical symbol \( \sqrt{} \). This form is useful for visualizing and understanding expressions that involve roots.
Converting an expression with a fractional exponent to radical form is straightforward. For \( \left(\frac{y}{x}\right)^{2/7} \), it translates to \( \sqrt[7]{\left(\frac{y}{x}\right)^2} \).

• The radical sign \( \sqrt{} \) indicates we're dealing with a root.
• The index of the radical (7 in this case) shows the degree of the root.
• What's underneath, \( \left(\frac{y}{x}\right)^2 \), is the expression being operated on.

This transformation is essential for rewriting and simplifying expressions, especially when communicating mathematical ideas.