Problem 84 - Algebra A Combined Function

Question

Given \(\frac{\sqrt[3]{5 y}}{\sqrt[3]{4}}\), rationalize the denominator by following parts (a) and (b). a. Multiply the numerator and denominator by \(\sqrt[3]{16}\). b. Multiply the numerator and denominator by \(\sqrt[3]{2}\). c. What can you conclude from parts (a) and (b)?

Step-by-Step Solution

Answer

Different multipliers in (a) and (b) rationalize the denominator leading to different numerators.

Step 1: Analyze the Original Expression

The original expression given is \(\frac{\sqrt[3]{5y}}{\sqrt[3]{4}}\). To rationalize the denominator, we aim to eliminate the cube root in the denominator.

Step 2: Part (a) - Multiply by \(\sqrt[3]{16}\)

Start by multiplying both the numerator and the denominator by \(\sqrt[3]{16}\): \[\frac{\sqrt[3]{5y}}{\sqrt[3]{4}} \times \frac{\sqrt[3]{16}}{\sqrt[3]{16}} = \frac{\sqrt[3]{80y}}{\sqrt[3]{64}}\]. Since \(\sqrt[3]{64} = 4\), the expression simplifies to \(\frac{\sqrt[3]{80y}}{4}\).

Step 3: Part (b) - Multiply by \(\sqrt[3]{2}\)

Now multiply both the numerator and the denominator by \(\sqrt[3]{2}\): \[\frac{\sqrt[3]{5y}}{\sqrt[3]{4}} \times \frac{\sqrt[3]{2}}{\sqrt[3]{2}} = \frac{\sqrt[3]{10y}}{\sqrt[3]{8}}\]. Since \(\sqrt[3]{8} = 2\), the expression simplifies to \(\frac{\sqrt[3]{10y}}{2}\).

Step 4: Compare the Results

In part (a), the simplified expression is \(\frac{\sqrt[3]{80y}}{4}\), and in part (b), it is \(\frac{\sqrt[3]{10y}}{2}\). Both parts have resulted in rationalized denominators, just with different numerators.

Step 5: Conclusion

Rationalizing the denominator can be achieved in different ways, resulting in different numerators, while the denominators become integers. The process showed how different choices of multiplication lead to the elimination of cube roots in the denominator.

Key Concepts

Understanding Cube Roots

A cube root is the number that you multiply by itself three times to get another number. For example, \(\sqrt[3]{8} = 2\)because \(2 imes 2 imes 2 = 8\). Cube roots are crucial when dealing with expressions that require rationalization, especially when the expression is under a root sign, like a cube root.

• The cube root notation \(\sqrt[3]{x}\) is used to denote the cube root of any number or variable \(x\).
• Cube roots are often involved in the process of simplifying fractions to have integer denominators.
• Rationalizing these expressions helps in standardizing measurements and comparing values without dealing with cube roots in the denominator.

In mathematics, simplifying expressions by rationalizing the denominator is common practice, making cube root concepts vital for understanding and manipulating algebraic expressions.

The Role of the Denominator in Rationalizing Expressions

The denominator is a key part of rationalizing expressions, especially when dealing with cube roots. It is the bottom part of the fraction that determines how many parts the whole is divided into. In expressions, ideally, the denominator should be a whole number instead of a root. This makes calculations easier and minimizes the potential for errors.

• When an expression involves cube roots in the denominator, it can complicate further calculations since cube roots can be challenging to work with as they are not always integers.
• To rationalize a denominator containing cube roots means replacing the root with an integer by multiplying the fraction by a form of 1 that eliminates the root.
• This process results in easier-to-read expressions and simplifies the arithmetic operations needed downstream.

Ultimately, the aim of dealing with the denominator in such expressions is to make the expression more workable and straightforward.

Algebraic Manipulation in Rationalizing Denominators

Algebraic manipulation involves adjusting expressions to make them more manageable or convey a different meaning while maintaining equality. When rationalizing denominators, especially those with cube roots, one key technique is using algebraic manipulation.

• By multiplying both the numerator and the denominator by a term that will eliminate the cube root, such as \(\sqrt[3]{16}\) or \(\sqrt[3]{2}\), you achieve a rational denominator.
• This method keeps the fraction equivalent (since multiplying by 1 doesn’t change its value) while simplifying computation by transforming cube roots into whole numbers.
• In our exercise, multiplying \(\frac{\sqrt[3]{5y}}{\sqrt[3]{4}}\) by \(\frac{\sqrt[3]{16}}{\sqrt[3]{16}}\) or \(\frac{\sqrt[3]{2}}{\sqrt[3]{2}}\) demonstrates how different algebraic approaches can simplify the expression without altering its original value.

These algebraic manipulations help create simplified expressions that are much easier to interpret and use, facilitating a clearer understanding of the overall mathematical concept being applied.