Problem 60 - Algebra A Combined Function

Question

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\sqrt[3]{\frac{9 y}{7}}\)

Step-by-Step Solution

Answer

Rationalized expression: \( \frac{3y \cdot \sqrt[3]{7}}{\sqrt[3]{21y^2}} \).

Step 1: Understand the Problem

The given expression to be rationalized is \( \sqrt[3]{\frac{9y}{7}} \). Rationalizing the numerator involves eliminating any radicals (roots) from the numerator of a fraction. Since this is a cube root, we'll manipulate it for rationalization.

Step 2: Identify Needed Factors

To rationalize, we need the numerator to be a perfect cube. The current numerator is 9y, expressed as the cube root: \( \sqrt[3]{9} \cdot \sqrt[3]{y} \). To make 9 a perfect cube, think of a factor that, when multiplied by 9, forms a perfect cube. We know that \(9 = 3^2\); therefore, multiplying by another \(3^1\) (which is \(3\)) makes it \(3^3 = 27\). For \(y\), multiplying by \(y^2\) will create \(y^3\). The needed factor for the entire numerator is \(3y^2\).

Step 3: Multiply the Numerator and Denominator

Multiply the entire expression by \( \frac{\sqrt[3]{3y^2}}{\sqrt[3]{3y^2}} \) to maintain the original value of the expression while rationalizing. This will give us: \[ \frac{\sqrt[3]{27y^3} \cdot \sqrt[3]{7}}{\sqrt[3]{21y^2}} \].

Step 4: Solve for Rationalized Numerator

Calculate the cube root of the numerator: \( \sqrt[3]{27y^3} = 3y \). Our new expression becomes: \( \frac{3y \cdot \sqrt[3]{7}}{\sqrt[3]{21y^2}} \). The numerator is now rationalized: 3y.

Key Concepts

Cube roots

A cube root, denoted by the symbol \( \sqrt[3]{} \), represents a number that, when multiplied by itself and then multiplied by itself again, equals the original number under the cube root. Simply put, if \( x^3 = a \), then \( x \) is the cube root of \( a \).Cube roots are crucial when dealing with expressions that involve radicals, as they help simplify terms by reducing them to their most basic form:

• Cube roots can appear in fractions, such as \( \sqrt[3]{\frac{9y}{7}} \).
• Solving a cube root expression often involves finding a simpler or more manageable form.
• Cube roots are particularly important when rationalizing numerators with radicals.

The goal when dealing with radicals, such as cube roots, is to simplify the expression to make it easier to understand and work with. Rationalizing, the process used in this exercise, helps achieve that simplicity by eliminating the radicals from the numerator.

Perfect cubes

A perfect cube is a number that can be expressed as the cube of an integer or variable: \( a^3 = b \). For example, 27 is a perfect cube because it can be expressed as \( 3 \times 3 \times 3 = 3^3 \).Understanding perfect cubes is helpful in:

• Simplifying radical expressions in mathematics by allowing counterparts of cube roots.
• Rationalizing numerators to achieve a simpler, rational form.

In the given exercise involving \( \sqrt[3]{\frac{9y}{7}} \), converting 9 into a perfect cube is necessary for rationalization. This involves multiplying by factors to make the original number (which is 9 in this case) a perfect cube, turning it into \( 3^3 = 27 \). By doing so, the cube root of 27 becomes 3, which is a whole number and contributes to a more straightforward expression when combined with other terms.

Fractions with radicals

When working with fractions that contain radicals, like \( \sqrt[3]{\frac{9y}{7}} \), the task often involves transforming the radical expression into a more manageable form. This concept, known as rationalization, focuses on removing the radicals from either the numerator or the denominator to simplify the fraction.To rationalize:

• Identify the necessary factors needed to make the radical expression a perfect cube.
• Multiply both the numerator and the denominator by these factors, in cube root form, to maintain the fraction's value.
• Simplify the expression by solving the cube root of the perfect cube in the numerator.

In practice, this process can seem complex, but it simplifies analytical work and calculations. After rationalizing the numerator in this exercise, we transformed \( \sqrt[3]{9y} \) into 3y, which is free of radicals, allowing for easier manipulation and understanding of the fraction.