Problem 51 - Algebra A Combined Function

Question

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{\sqrt[3]{5 y^{2}}}{\sqrt[3]{4 x}}\)

Step-by-Step Solution

Answer

The rationalized expression is \(\frac{5y^2}{\sqrt[3]{100xy^4}}\).

Step 1: Identify the Need to Rationalize

To rationalize the numerator of the expression \(\frac{\sqrt[3]{5 y^{2}}}{\sqrt[3]{4 x}}\), we need to eliminate the cube root in the numerator. This is done by multiplying the numerator and the denominator by a term that makes the expression a perfect cube.

Step 2: Determine the Rationalizing Factor

Since the numerator is \(\sqrt[3]{5 y^2}\), to make it a perfect cube, we need another factor of \(5y^2\). Therefore, the rationalizing factor is \(5^{2/3} y^{4/3}\).

Step 3: Multiply to Eliminate the Cube Root

Multiply both the numerator and the denominator by \(\sqrt[3]{25 y^4}\) (which is the decimal representation of \((5y^2)^{2/3}\)): \[\frac{\sqrt[3]{5 y^2} \cdot \sqrt[3]{25 y^4}}{\sqrt[3]{4 x} \cdot \sqrt[3]{25 y^4}}\] This simplifies to \[\frac{\sqrt[3]{(5 y^2)^3}}{\sqrt[3]{100 x y^4}}\]

Step 4: Simplify the Expression

The cube root of the numerator, where the expression is \((5y^2)^3\), becomes \(5y^2\). Therefore, the numerator simplifies to 5y^2. The denominator becomes \(\sqrt[3]{100x y^4}\).

Key Concepts

Rationalization Process

Rationalization is a technique used in mathematics to simplify expressions involving roots. In our problem, this involves removing the cube root from the numerator.

• This is done by multiplying both the numerator and denominator by a strategic value, called the rationalizing factor.
• By making the numerator a perfect cube, we eliminate the root without altering the overall value of the fraction.

This is crucial in expressions, as it often results in a cleaner, more manageable form. Imagine rationalization as bringing order to chaos—a handy trick for transforming complex fractional expressions into simpler ones.

Cube Roots

A cube root asks what number, multiplied by itself three times, gives us the original value.

• For instance, the cube root of 8 is 2, because 2 multiplied by itself thrice is 8.
• In cube root expressions, we often need to adjust them to reach a simpler form.

In our task, the numerator is \(\sqrt[3]{5y^2}\), which represents a less tidy form that we wish to clean up through rationalization. By multiplying the expression in such a way that our numerator becomes a perfect cube, the root vanishes, transforming it into a more straightforward numerical value.

Mathematical Expressions

Mathematical expressions like the one given in the problem can at first seem a bit daunting, but they are simply representations of numbers and operations performed on them.

• In our particular case, \(\frac{\sqrt[3]{5y^2}}{\sqrt[3]{4x}}\), we have a division of two cube root expressions.
• Understanding these expressions requires knowing the interplay of numbers, roots, and operations.

The goal is to manipulate these expressions in a way that simplifies them while maintaining their equality. By focusing on the roots and applying orderly mathematical operations, we simplify complex problems efficiently.

Multiplying Rational Expressions

Multiplying rational expressions is a key part of the rationalization process.

• This involves combining fractions by multiplying both numerators and denominators.
• Let's consider our example: we multiply the expression by \(\sqrt[3]{25y^4}\) on both top and bottom.

The trick is to choose a multiplying factor that simplifies the expression.By doing this, we exploit the properties of exponents and roots to cancel the cube root in the numerator, resulting in a rational expression. Understanding how to choose these multipliers and what they achieve is integral to mastering multiplying rational expressions.