Problem 10 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{5}{\sqrt[3]{3 y}}\)

Step-by-Step Solution

Answer

\( \frac{5 (\sqrt[3]{3y})^2}{3y} \)

Step 1: Identify the Denominator

First, identify the denominator that needs to be rationalized. In this case, it's \( \sqrt[3]{3y} \), a cube root.

Step 2: Determine the Multiplicative Factor

To rationalize the denominator \( \sqrt[3]{3y} \), we need to multiply the fraction by a form that will remove the cubed root. This means multiplying by \( \left( \sqrt[3]{3y} \right)^2 \) since \( \left( \sqrt[3]{3y} \right)^3 = 3y \).

Step 3: Multiply Numerator and Denominator by the Factor

Multiply both the numerator and the denominator by \( \left( \sqrt[3]{3y} \right)^2 \): \[ \frac{5}{\sqrt[3]{3y}} \times \frac{(\sqrt[3]{3y})^2}{(\sqrt[3]{3y})^2} = \frac{5(\sqrt[3]{3y})^2}{\sqrt[3]{3y} \cdot (\sqrt[3]{3y})^2} \]

Step 4: Simplify the Denominator

Simplify the denominator: \[ \sqrt[3]{3y} \cdot (\sqrt[3]{3y})^2 = (\sqrt[3]{3y})^3 = 3y \] This results in a rational number.

Step 5: Simplify the Fraction

Now, simplify the fraction: \[ \frac{5(\sqrt[3]{3y})^2}{3y} \]We don't change the numerator, so the final expression is \[ \frac{5 \cdot (\sqrt[3]{3y})^2}{3y} \]

Key Concepts

Rationalization

Rationalization is a powerful algebraic technique used to eliminate irrationality from the denominator of a fraction. As its name suggests, it transforms an irrational expression into a rational one. This process often involves multiplying both the numerator and denominator by a term that will cancel out the unwieldy roots or radicals. The primary goal is to ensure the denominator becomes a perfect power of the numbers involved.
While rationalization is straightforward for square roots, dealing with cube roots or higher requires identifying the appropriate multiplicative factor. In the given exercise, this involves multiplying by a power that raises a cube root to a whole number, eliminating it.
Remember, multiplying the fraction by 1 in a strategic form ensures the value remains unchanged even as its form transforms.

Cube Roots

Cube roots are the set members of radicals and denote the number which, when multiplied by itself three times, gives the original number.
For instance, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\) or expressed as \(\sqrt[3]{8} = 2\).
When dealing with cube roots, understanding exponentiation is crucial.

• The cube root of a number can be represented as that number raised to the power of \(1/3\).
• To rationalize an expression containing cube roots, you may need to multiply by the square of the cube root to make the exponent a full integer, as shown in the solution.

By multiplying the radical expression effectively, you convert the cube root into a clear, whole number by making \((\sqrt[3]{3y})^3 = 3y\), hence simplifying operations.

Fractions

Fractions express the division of one quantity by another and are a cornerstone of algebra. Communicating relationships between numbers through division, fractions might incorporate radical expressions, as seen in this exercise.
When presented with a fraction where the denominator contains radicals, it's essential to simplify to enhance clarity and calculations.

• Maintain equality in the expression by multiplying both numerator and denominator by the same value.
• Crafting a strategy around adjusting the fraction's form, not its value, follows the principles of mathematical balance.
  For instance, multiplying \(\frac{5}{\sqrt[3]{3y}}\) by \((\sqrt[3]{3y})^2/(\sqrt[3]{3y})^2\) reframes the expression while preserving its original value.

Denominator Simplification

Denominator simplification aims to tidy and clarify expressions by eliminating radicals or reducing terms to basic, manageable numbers.

• The primary method in simplifying denominators such as \(\sqrt[3]{3y}\) involves using powers to equate any irrational segment to an integer.
• In our case, multiplying by \((\sqrt[3]{3y})^2\) transforms the denominator into \(3y\), uncomplicating the expression.

By making the denominator rational, you ensure formulations are neat and straightforward, allowing further operations, such as addition or subtraction of fractions, to proceed without complexity barriers.
This simplification doesn't just adhere to mathematical principles but also aids in practical problem-solving, ensuring clarity in communication and analysis.