Problem 52 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The rationalized expression is \(\frac{4x}{\sqrt[3]{16x^2z^4}}\).

Step 1: Understand the Problem

The problem requires us to rationalize the numerator of the fraction \(\frac{\sqrt[3]{4x}}{\sqrt[3]{z^4}}\). Rationalizing the numerator means eliminating the cube root in the numerator and expressing it in a simpler, rational form.

Step 2: Multiply by a Conjugate Factor

To rationalize a cube root in the numerator, we can multiply both the numerator and the denominator by the cube root that will make the numerator a complete power of the cube root base. In this case, multiply by \(\sqrt[3]{(4x)^2}\):\[\frac{\sqrt[3]{4x} \cdot \sqrt[3]{(4x)^2}}{\sqrt[3]{z^4} \cdot \sqrt[3]{(4x)^2}}\]

Step 3: Simplify the Numerator

Now, simplify the result in the numerator. Using the property \(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab}\), we have:\[\sqrt[3]{4x \cdot (4x)^2} = \sqrt[3]{4^3 x^3} = |4x| = 4x\]Since all variables and constants are positive, the absolute value can be omitted, leaving us with \(4x\).

Step 4: Simplify the Denominator

Similarly, simplify the denominator:\[(\sqrt[3]{z^4} \cdot \sqrt[3]{(4x)^2}) = \sqrt[3]{z^4 \cdot (16x^2)}\]This simplifies to \(\sqrt[3]{16x^2z^4}\).

Step 5: Final Expression

We've successfully rationalized the numerator and now have:\[\frac{4x}{\sqrt[3]{16x^2z^4}}\]This is the rationalized expression.

Key Concepts

Cube Roots

Cube roots are a type of radical expression used to find a number that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because when you multiply 2 by itself three times (2 \( \times \) 2 \( \times \) 2), you get 8.
In mathematical terms, the cube root of a number \( a \) is represented as \( \sqrt[3]{a} \). This is known as a radial expression and is commonly used in various mathematical calculations.

Cube roots have key properties that are used in simplification processes:

• Cube root of a product: \( \sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b} \).
• Cube root of a power: If \( a^3 = b \), then \( a = \sqrt[3]{b} \).

In the context of rationalizing numerators, we often multiply by another term so that the result under the cube root is a perfect power (like \( 4x \) in the solution) which can then be simplified further.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. These are an essential part of algebra and are often found in ratios, rates, and expressions.
Working with rational expressions requires manipulation to achieve simplification. This is because it often includes radical components, like cube roots, that need rationalizing.

In terms of rationalizing a numerator, the goal is to remove radicals by multiplying by an appropriate factor. The rational expression can usually be rewritten by:

• Multiplying the numerator and denominator by a conjugate or necessary factor to eliminate denominators or numerators with radicals.
• Rewriting the expression in a simpler form to remove radicals, allowing easier evaluation or further algebraic manipulation.

Rational expressions combine many algebraic concepts, offering a foundation for more advanced topics.

Mathematical Simplification

Mathematical simplification involves rewriting an expression in a simpler form while retaining its value. Simplification is key for easier computation and understanding of complex expressions.
Simplifying rational expressions, particularly those with radicals like cube roots, involves a few helpful techniques:

• Combine like terms or expressions using their properties (e.g., \( \sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \cdot b} \)).
• Identify and cancel common factors where possible.
• Rationalizing numerators or denominators by eliminating roots using appropriate multiplication, resulting in simpler, complete power forms.

Ultimately, simplification allows us to deal with less complex expressions, making mathematical operations and problem-solving more efficient. It forms a central part of learning in not just algebra but across all areas of mathematics.