Problem 71 - Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{\sqrt[3]{12 t^{3}}}{\sqrt[3]{54 t^{2}}} $$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{\sqrt[3]{6t}}{3} \).

Step 1: Simplify the Radicals

Start by simplifying each of the cube roots. For the numerator \( \sqrt[3]{12t^3} \), we can express 12 as \( 2^2 \times 3 \), so we have: \[ \sqrt[3]{12t^3} = \sqrt[3]{2^2 \times 3 \times t^3} = 2 \times t \times \sqrt[3]{3} \]. For the denominator \( \sqrt[3]{54t^2} \), express 54 as \( 2 \times 3^3 \): \[ \sqrt[3]{54t^2} = \sqrt[3]{2 \times 3^3 \times t^2} = 3 \times \sqrt[3]{2t^2} \].

Step 2: Form the Fraction

Now form the fraction using the simplified radicals you obtained:\[ \frac{2t \sqrt[3]{3}}{3 \sqrt[3]{2t^2}} \].

Step 3: Rationalize the Denominator

To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt[3]{2t} \) so that the denominator becomes a complete cube. This gives us:\[ \frac{2t \sqrt[3]{3} \times \sqrt[3]{2t}}{3 \sqrt[3]{2t^2} \times \sqrt[3]{2t}} = \frac{2t \sqrt[3]{6t}}{3 \times 2t} = \frac{2t \sqrt[3]{6t}}{6t} \].

Step 4: Simplify the Expression

Since \( 2t \) in the numerator and denominator cancel each other out, simplify the fraction: \[ \frac{\sqrt[3]{6t}}{3} \]. This is the fully rationalized form.

Key Concepts

Cube Roots

Cube roots are mathematical expressions that find a number which, when multiplied by itself three times, yields the original number. For instance, the cube root of 8 is 2, because 2 x 2 x 2 = 8.
When working with numbers inside a cube root, it is helpful to break them down into their prime factors. This makes it easier to identify any perfect cube factors that can be simplified out of the radical.

• Understanding Cube Roots: They are denoted by the radical sign with a small three (\( \sqrt[3]{ } \)).
• Prime Factorization: Breaking numbers down to their prime factors allows recognition of perfect cubes (e.g., 27 = 3 x 3 x 3, making \( \sqrt[3]{27} = 3 \)).
• Variable Considerations: Variables raised to multiples of three can be simplified similarly (e.g., \( \sqrt[3]{t^3} = t \)).

Simplifying Radicals

Simplifying radicals involves breaking down a number or expression under a radical sign into its simplest form. This often involves factoring out perfect squares, cubes, or other powers, depending on the index of the radical.

• Why Simplify Radicals: It makes expressions easier to work with and is often a necessary step in problem-solving, especially when rationalizing denominators.
• Process: Begin by factoring the number or expression into its prime factors and group them according to the radical's index. For cube roots, look for groups of three identical factors.
• Example: In the given problem, \( \sqrt[3]{12t^3} \) simplifies to \( 2t \sqrt[3]{3} \) by identifying \( t^3 \) as a perfect cube and factoring 12 into \( 2^2 \times 3 \).

Positive Real Numbers

When dealing with cube roots and radicals, especially in rationalizing denominators, it is essential to assume that all variables represent positive real numbers. This allows for simplification without worrying about negative roots or imaginary numbers, which could complicate the process.

• Definition: A positive real number is any real number greater than zero. In radical expressions, the assumption of positivity removes ambiguity related to the square and cube roots of variables.
• Consistency: By assuming all variables are positive, expressions like \( \sqrt[3]{t^3} = t \) are valid without additional conditions on \( t \).
• Implications for Simplification: This assumption means that when variables cancel out in the process of simplification, the results remain valid and true to their mathematical meaning.

By understanding these core concepts, tackling problems that involve rationalizing denominators becomes a more straightforward process, and students can better handle the simplification of cube roots and radicals efficiently.