Problem 70 - Precalculus: Functions and Graphs

Question

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt[3]{\frac{3 x^{2} y^{5}}{4 x}}$$

Step-by-Step Solution

Answer

The simplified and rationalized expression is \( \frac{y \sqrt[3]{48xy^2}}{4} \).

Step 1: Simplify the Fraction Inside the Radical

First, simplify the fraction inside the cube root. The expression inside the radical is \( \frac{3x^2y^5}{4x} \). This can be simplified by canceling the common term \( x \) from the numerator and the denominator.After canceling, we have: \[ \frac{3x^2y^5}{4x} = \frac{3xy^5}{4} \]

Step 2: Apply the Cube Root to the Numerator and Denominator

Apply the cube root to both the numerator and the denominator separately:\[ \sqrt[3]{\frac{3xy^5}{4}} = \frac{\sqrt[3]{3xy^5}}{\sqrt[3]{4}} \]

Step 3: Simplify the Cube Root of the Numerator

Now, simplify \( \sqrt[3]{3xy^5} \). Notice that \( y^5 \) can be rewritten as \( y^3 \times y^2 \).So it can be expressed as:\[ \sqrt[3]{3xy^5} = \sqrt[3]{3xy^3y^2} = y \sqrt[3]{3xy^2} \]

Step 4: Simplify the Cube Root of the Denominator

Simplify \( \sqrt[3]{4} \). Since 4 is \( 2^2 \), there is no perfect cube in the denominator.Thus, it remains \( \sqrt[3]{4} \).

Step 5: Rationalize the Denominator

To rationalize the denominator, multiply the numerator and the denominator by \( \sqrt[3]{4^2} = \sqrt[3]{16} \). Thus:\[ \frac{y \sqrt[3]{3xy^2}}{\sqrt[3]{4}} \cdot \frac{\sqrt[3]{16}}{\sqrt[3]{16}} = \frac{y \sqrt[3]{48xy^2}}{4} \]

Step 6: Simplified Expression

After rationalizing, the expression becomes:\[ \frac{y \sqrt[3]{48xy^2}}{4} \]This is the simplified and rationalized expression.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate radicals from the denominator of a fraction, making it easier to understand or work with. When the denominator contains \(\sqrt[3]{4}\), for instance, it cannot be simplified directly since 4 is not a perfect cube. Instead, we should aim to make the denominator a cube by multiplying both the numerator and the denominator by a term that will yield a perfect cube.

In our exercise, the denominator is \(\sqrt[3]{4}\), which can be rationalized by multiplying by \(\sqrt[3]{4^2} = \sqrt[3]{16}\) since \(4^3 = 64\), which is a perfect cube. Remember that any operation on the denominator must also be performed on the numerator to keep the overall value of the expression unchanged.

• Multiply the numerator by \(\sqrt[3]{16}\)
• Multiply the denominator by \(\sqrt[3]{16}\)

This results in eliminating the cube root from the denominator, as multiplying these together yields a perfect cube.

Cube Roots

Understanding cube roots is essential to solving expressions involving radical simplification, such as the one in this exercise. A cube root of a number \(a\) is a value \(b\) such that \(b^3 = a\). For instance, the cube root of 8 is 2 because \(2^3 = 8\).

When simplifying the cube root of an expression, it is helpful to identify factors of terms that can form perfect cubes. In our case, to simplify \(\sqrt[3]{3xy^5}\), we recognize that \(y^5\) can be broken into \(y^3 \times y^2\). Thus, \(\sqrt[3]{y^3} = y\). The remaining expression inside the cube root becomes \(\sqrt[3]{3xy^2}\).

• Identify perfect cube factors
• Extract the cube roots of perfect cube factors
• Leave non-cubic terms inside the radical

This step allows us to partially simplify the cube root to further refine the radical expression.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their most reduced or most straightforward form. This is done to make them easier to interpret or solve in algebraic operations.

In the provided exercise, we first simplified the expression \(\frac{3x^2y^5}{4x}\), by removing the common term \(x\) present in both the numerator and denominator. This leaves us with a simpler fraction: \(\frac{3xy^5}{4}\). From here, each step follows logically to simplify and, if necessary, rationalize the expression further.

• Cancel out common terms
• Simplify fractions by reducing terms
• Use algebraic identities for simplification

These processes allow ambitious algebraic expressions to transform into manageable forms. Algebraic simplification is typically the first step before further operations, such as rationalizing denominators or simplifying cube roots.