Problem 70 - Elementary and Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{\sqrt[3]{15 m^{4}}}{\sqrt[3]{12 m^{3}}} $$

Step-by-Step Solution

Answer

\( \sqrt[3]{\frac{5m}{4}} \) is the rationalized expression.

Step 1: Simplify the Fraction

Start by rewriting the given fraction \( \frac{\sqrt[3]{15 m^{4}}}{\sqrt[3]{12 m^{3}}} \) as a single cube root: \( \sqrt[3]{\frac{15 m^{4}}{12 m^{3}}} \).

Step 2: Simplify Inside the Cube Root

Simplify the expression inside the cube root: \( \frac{15 m^{4}}{12 m^{3}} = \frac{15}{12} \cdot \frac{m^{4}}{m^{3}} = \frac{5}{4} \cdot m \).

Step 3: Rationalize the Denominator

To rationalize the denominator, which is a cube root, you want to make it a perfect cube. Multiply by \( \sqrt[3]{\frac{4^2}{4^2}} = \sqrt[3]{\frac{16}{16}} \) to clear the cube root in the denominator.

Step 4: Perform Multiplication

Now, multiply: \( \sqrt[3]{\frac{5}{4} \cdot m} \cdot \sqrt[3]{\frac{16}{16}} = \sqrt[3]{\frac{5 \times 16 \times m}{4^3}} = \sqrt[3]{\frac{80m}{64}} \).

Step 5: Simplify the Rationalized Expression

Simplify the result inside the cube root: \( \frac{80m}{64} = \frac{5m}{4} \), resulting in \( \sqrt[3]{\frac{5m}{4}} \).

Key Concepts

Cube Root Simplification

Understanding cube root simplification is a powerful tool when working with more complex algebraic expressions. The cube root of a number finds which number, when multiplied by itself twice, will equal the original number. Simplifying cube roots involves breaking down the number or the expression under the root into smaller parts that are easier to work with.

In the given equation, the expression under the cube root sign, \( \sqrt[3]{\frac{15 m^4}{12 m^3}} \), can first be broken down into separate components using division inside the root. Calculating the fraction \( \frac{15}{12} \) boils down to its simplest form, \( \frac{5}{4} \), and simplifying the variable part \( \frac{m^4}{m^3} \) gives you just \( m \). Thus, the initial expression changes into \( \sqrt[3]{\frac{5}{4} \cdot m} \).

We continue this simplification by ensuring any resulting terms under the cube root are as simple as they possibly can be, often preparing them for further procedures like rationalizing denominators.

Fraction Simplification

Fraction simplification is about expressing a fraction in its simplest form, which involves reducing both the numerator and the denominator to their smallest possible whole numbers. This makes calculations more straightforward and results clearer.

To simplify a fraction, break down both the numerator and the denominator into their prime factors and cancel out any common factors they share. For the fraction \( \frac{15}{12} \), this involves dividing both by their greatest common divisor, which is 3, resulting in \( \frac{5}{4} \).

This reduction is a critical step in the process to ensure a clean path toward rationalizing the denominator or simplifying expressions under cube roots. Simplifying eliminates unnecessary complexity and makes tackling subsequent mathematical operations more seamless.

Algebraic Expressions

Algebraic expressions are a fundamental aspect of algebra and mathematics as a whole. They typically include numbers, variables, and arithmetic operations. Proper manipulation and simplification of these expressions are critical in solving more complex problems.

Expressions involving roots, like cube roots, can often be simplified by using exponent rules and property of fractions. In the context of rationalizing denominators, the goal is to eliminate any roots that exist in the denominator, transforming it into a whole number without changing the value of the expression.

In our problem: \( \frac{\sqrt[3]{15 m^4}}{\sqrt[3]{12 m^3}} \), we applied algebraic principles to simplify and make the expression more manageable. This included the division of the terms and using multiplication to eliminate roots from the denominator. Overall, understanding and working efficiently with algebraic expressions boosts mathematical proficiency significantly, aiding in clear and accurate simplification.