Problem 42 - Algebra 2

Question

Simplify. Rationalize all denominators. Assume that all the variables are positive. $$ \frac{3+\sqrt[3]{2}}{\sqrt[3]{2}} $$

Step-by-Step Solution

Answer

The rationalized and simplified form of the given expression is \((1-2\sqrt[3]{2})\) over \(2-\sqrt[3]{4}\)

Step 1: Identify the Expression to be Rationalized

The expression given to us is \(\frac{3+\sqrt[3]{2}}{\sqrt[3]{2}}\). Here, \(\sqrt[3]{2}\) in the denominator is the term that we need to rationalize.

Step 2: Use the Difference of Cubes Formula

To rationalize the denominator, we need to use the difference of cubes formula which states that \(a^3-b^3 = (a-b)(a^2+ab+b^2)\). Renaming \(\sqrt[3]{2}\) as 'a' and 1 as 'b', we aim to obtain a term equivalent to \(a^3 - b^3\) which we can then simplify.

Step 3: Rationalize the Denominator

To achieve the term equivalent to \(a^3 - b^3\), we multiply the numerator and denominator by \(1-\sqrt[3]{2}\). So the expression becomes \((3+\sqrt[3]{2})(1-\sqrt[3]{2})\) over \((\sqrt[3]{2})(1-\sqrt[3]{2})\)

Step 4: Simplify the Expression

After multiplication, we obtain \((3-3\sqrt[3]{2}+\sqrt[3]{2}-2)\) over \((2-\sqrt[3]{4})\). Simplifying further, we get \((1-2\sqrt[3]{2})\) over \(2-\sqrt[3]{4}\)

Key Concepts

Radicals

Radicals are expressions that contain a root symbol, like the square root or cube root. The radical expression in the problem is \( \sqrt[3]{2} \), which represents the cube root of 2. Radicals can sometimes make equations more complicated, especially when they appear in denominators. To simplify math expressions, we often need to "rationalize" them—meaning we convert the expression into an equivalent one with a rational denominator. This involves getting rid of any roots in the denominator, which can be achieved with algebraic manipulation, often using identities or formulas such as those for the difference of cubes.

Difference of Cubes

The difference of cubes formula is \[ a^3 - b^3 = (a-b)(a^2+ab+b^2) \] This formula is particularly useful when dealing with cube roots, as it helps us eliminate cube roots in the expression. For instance, in our exercise, we use \( a = \sqrt[3]{2} \) and \( b = 1 \) to apply the difference of cubes.By skillfully choosing \( a \) and \( b \), and recognizing our cube roots as a part of this formula, we can express products as cubes, allowing us to rationalize these expressions. This method streamlines the task of simplifying the original form.

Denominator

A denominator is the bottom part of a fraction which tells us how many parts the whole is divided into. In the problem, the denominator \( \sqrt[3]{2} \) makes the fraction irrational. To handle this, we aim to eliminate the radical from the denominator by making it a rational number. We achieve this by multiplying the numerator and the denominator by a specific term, which in this case is \( (1-\sqrt[3]{2}) \).This step is crucial because having a rational denominator simplifies further calculations and gives clearer results. It's not just about simplification, but also about ease in future operations with these expressions.

Simplification

Simplification involves transforming an expression into its simplest or most easily manageable form. For our problem, once we've rationalized the denominator, we multiply out the terms. In the numerator, we have \( (3+\sqrt[3]{2})(1-\sqrt[3]{2}) \) Expanding this gives us: - \( 3 \times 1 = 3 \)- \( 3 \times -\sqrt[3]{2} = -3\sqrt[3]{2} \)- \( \sqrt[3]{2} \times 1 = \sqrt[3]{2} \)- \( \sqrt[3]{2} \times -\sqrt[3]{2} = -2 \)Combining all, the expression simplifies to \( 1 - 2\sqrt[3]{2} \) in the numerator.In the denominator, multiplying \( (\sqrt[3]{2})(1-\sqrt[3]{2}) \) and simplifying yields \( 2-\sqrt[3]{4} \). This simplification gives us a cleaner, more comprehensible expression that reveals the hidden simplicity of the original problem. It's about reducing complexity while maintaining equality.