Rationalizing the Denominator of an Expression Involving Cube Roots

In mathematics, rationalizing the denominator is an important technique used to simplify expressions especially when dealing with fractions containing radicals or roots in the denominator. This guide will walk you through the process of rationalizing the denominator for the expression ( frac{1}{2^{1/3} 2^{-1/3}} ). We will use this opportunity to explore algebraic manipulations and demonstrate a step-by-step approach.

Step 1: Understanding the Expression

Let's begin by understanding our initial expression:

Expression: ( frac{1}{2^{1/3} 2^{-1/3}} )

Note that: ( 2^{-1/3} frac{1}{2^{1/3}} )

Rewrite the expression for clarity:

( frac{1}{2^{1/3} 2^{-1/3}} frac{1}{2^{1/3} frac{1}{2^{1/3}}} )

Step 2: Finding the Conjugate

The conjugate of an expression involving cube roots can be found similarly to how it's done with square roots. However, for cube roots, the conjugate will help in eliminating the radicals in the denominator. The conjugate of ( 2^{1/3} 2^{-1/3} ) is ( 2^{1/3} - 2^{-1/3} ).

Multiply the numerator and the denominator by the conjugate:

( frac{1}{2^{1/3} 2^{-1/3}} cdot frac{2^{1/3} - 2^{-1/3}}{2^{1/3} - 2^{-1/3}} frac{2^{1/3} - 2^{-1/3}}{2^{1/3} frac{1}{2^{1/3}} (2^{1/3} - 2^{-1/3})} )

Step 3: Simplifying the Denominator

The denominator can be simplified using the difference of squares formula:

( (2^{1/3} 2^{-1/3})(2^{1/3} - 2^{-1/3}) 2^{2/3} - 2^{-2/3} )

Further simplification of the expression:

( 2^{2/3} - 2^{-2/3} 2^{2/3} - frac{1}{2^{2/3}} )

( frac{2^{4/3} - 1}{2^{2/3}} )

Substituting back into the expression:

( frac{2^{1/3} - 2^{-1/3}}{frac{2^{4/3} - 1}{2^{2/3}}} frac{2^{1/3} - 2^{-1/3} 2^{2/3}}{2^{4/3} - 1} )

The numerator simplifies to:

( 2^{1/3} 2^{2/3} - 2^{-1/3} 2^{2/3} 2 - 2^{1/3} )

Final simplified form:

( frac{2 - 2^{1/3}}{2^{4/3} - 1} )

Step 4: Simplifying Further (Alternative Method)

Another approach involves using the identity ( a^3 - b^3 (a - b)(a^2 ab b^2) ). Let's set ( a 2^{1/3} ) and ( b 2^{-1/3} ).

Multiply the numerator and the denominator by the conjugate:

( frac{1}{2^{1/3} 2^{-1/3}} frac{1}{2^{1/3} 2^{-1/3}} cdot frac{2^{2/3} - 2^{1/3}2^{-1/3} 2^{-2/3}}{2^{2/3} - 2^{1/3}2^{-1/3} 2^{-2/3}} )

Simplify the denominator:

( 2^{1/3} 2^{-1/3} 2^{1/3 - 1/3} 2^0 1 )

( 2^{2/3} - 2^{1/3}2^{-1/3} 2^{-2/3} 2^{2/3} - (2^{1/3 - 1/3} 2^{-2/3}) 2^{2/3} - frac{1}{2^{2/3}} frac{2^{4/3} - 1}{2^{2/3}} )

This further simplifies to:

( frac{1}{2^{1/3} 2^{-1/3}} frac{2^{1/3} 2^{2/3} - 2^{-1/3} 2^{2/3}}{2 1} frac{2 - 2^{1/3}}{2^{4/3} - 1} )

Step 5: Final Rationalized Form

Combining the results, we have:

( frac{2 - 2^{1/3}}{2^{4/3} - 1} )

This is the final rationalized form of the given expression.

Additional Notes and Tips

Rationalizing the denominator is crucial in several mathematical operations, especially when dealing with fractions involving radicals. It helps in simplifying expressions and making them more manageable for further calculations. Remember that the key is to use the appropriate algebraic identities and properties of radicals to achieve the desired outcome.

For more practice and a deeper understanding, consider exploring similar expressions and applying these techniques. Understanding and mastering these concepts can significantly improve your algebraic skills and mathematical fluency.