Problem 70 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: 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

Let's start by simplifying the fraction inside the cube root. The expression is \( \frac{3x^2y^5}{4x} \). Divide the numerator and the denominator by \( x \). This gives us \( \frac{3xy^5}{4} \).

Step 2: Apply the Cube Root to the Fraction

Now, apply the cube root to both the numerator and the denominator separately: \( \sqrt[3]{3xy^5} \) and \( \sqrt[3]{4} \).

Step 3: Simplify the Numerator

To simplify \( \sqrt[3]{3xy^5} \), consider each part separately. The \( 3 \) and \( x \) remain under the cube root, but \( y^5 \) can be written as \( (y^3)\cdot(y^2) \). Thus, \( \sqrt[3]{y^5} = y \sqrt[3]{y^2} \).

Step 4: Simplify the Expression

Combine the results: \( \sqrt[3]{3xy^5} = y \sqrt[3]{3xy^2} \). Therefore, the entire expression becomes \( \frac{y \sqrt[3]{3xy^2}}{\sqrt[3]{4}} \).

Step 5: Rationalize the Denominator

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

Key Concepts

Cube Root Simplification

When dealing with cube roots, it's essential to break down expressions in a way that allows for simplification. A cube root, denoted by \(\sqrt[3]{\cdot}\), finds a value that, when multiplied by itself three times, results in the original number. Simplifying a cube root involves factoring the terms inside. For example, if given \(\sqrt[3]{y^5}\), the exponent of 5 can be broken down into \((y^3)\cdot(y^2)\).

• The cube root of \(y^3\) is \(y\) because \(y \cdot y \cdot y = y^3\).
• The \(y^2\) remains under the cube root as it cannot be simplified further.

So, \(\sqrt[3]{y^5} = y \sqrt[3]{y^2}\). This kind of breakdown into smaller exponent parts makes it easier to simplify expressions with higher exponents under cube roots.

Exponents in Fractions

Exponents in fractions can initially seem daunting, but they simplify through division and applying fundamental rules of exponents. When you see an expression like \(\frac{x^a}{x^b}\), you can simplify it using the rule \(x^{a-b}\). In the example, \(\frac{3x^2y^5}{4x}\), when we divide the \(x^2\) by \(x\), it becomes \(x^{2-1} = x\).

• The exponents are subtracted because of the division.
• Remember to only apply this rule when both terms have the same base.

Through simplifying exponents in fractions, complex expressions become more manageable, assisting in reducing expressions to their simplest forms.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations, making them a vital part of mathematics. Simplifying an algebraic expression requires applying the order of operations and properly managing different components like variables and constants. Consider \(\frac{3xy^5}{4}\), with different variables and opérations.

• Understand how each term interacts within an expression.
• Identify similar terms that can be combined or simplified.

Variables like \(x\) and \(y^5\) represent unknown quantities and require manipulation to align with the rest of the expression. Simplifying expressions entails applying appropriate mathematical operations, ensuring the final result is as succinct as possible. Whether it's cancelling terms or adjusting exponents, understanding algebraic expressions is crucial for handling more complex equations.