Problem 53 - Contemporary Precalculus

Question

Use the fact that \(x^{3}+y^{3}=\) \((x+y)\left(x^{2}-x y+y^{2}\right)\) to rationalize the denominator. $$\frac{1}{\sqrt[3]{3}+1}$$

Step-by-Step Solution

Answer

Question: Rationalize the denominator of the fraction $$\frac{1}{\sqrt[3]{3}+1}$$ using the given identity: \(x^{3}+y^{3}=(x+y)(x^{2}-x y+y^{2})\). Answer: The rationalized form of the given fraction is $$\frac{3-\sqrt[3]{3}+1}{4}.$$

Step 1: Identify x and y

In this case, we must choose the values of x and y to match the denominator: \(\sqrt[3]{3} + 1\). So we pick \(x = \sqrt[3]{3}\) and \(y = 1\).

Step 2: Apply the given identity

Substitute the values of \(x\) and \(y\) into the given identity: $$ x^{3}+y^{3}=(\sqrt[3]{3}+1)\left((\sqrt[3]{3})^{2}-\sqrt[3]{3}+1^{2}\right) $$

Step 3: Simplify the given identity

Simplify the expression: $$ \sqrt[3]{3^3}+1^{3}=(\sqrt[3]{3}+1)\left(3-\sqrt[3]{3}+1\right) $$

Step 4: Further simplification

Simplify the expression even more by removing the cube root and evaluate the sum: $$ 3 + 1 = (\sqrt[3]{3}+1)(3-\sqrt[3]{3}+1) $$

Step 5: Find the conjugate

We need to find a conjugate of the denominator to rationalize it. To do so, we multiply both the numerator and the denominator of the given fraction by the conjugate of the denominator, which is \(3-\sqrt[3]{3}+1\). $$ \frac{1}{\sqrt[3]{3}+1} \cdot \frac{3-\sqrt[3]{3}+1}{3-\sqrt[3]{3}+1} $$

Step 6: Multiply and simplify

Now, multiply the numerators and the denominators, and use the result from Step 4 to simplify the expression: $$ \frac{3-\sqrt[3]{3}+1}{(\sqrt[3]{3}+1)(3-\sqrt[3]{3}+1)} = \frac{3-\sqrt[3]{3}+1}{3+1} $$

Step 7: Final simplification

Simplify the fraction and this will be our final answer: $$ \frac{3-\sqrt[3]{3}+1}{4} $$

Key Concepts

Cubic Roots

When we talk about cubic roots, we are referring to the number that, when multiplied by itself three times, gives the original number. For instance, if we have the cubic root of 27, it equals 3 because 3 multiplied by itself three times (i.e., 3 \( \times \) 3 \( \times \) 3) equals 27. Cubic roots are represented as \( \sqrt[3]{\cdot} \), which signifies that you are taking the cube root of a number.
In this exercise, we are dealing with \( \sqrt[3]{3} \), which indicates the cube root of 3.

• To compute this manually, you would search for a number whose third power is close to or equals 3.
• Cube roots can sometimes be approximate due to irrational numbers, which cannot be precisely expressed as a fraction or a decimal.

Understanding this concept is important because it helps us to simplify expressions involving roots, particularly when we try to rationalize denominators, as seen in the given exercise.

Factoring Identities

Factoring identities are special algebraic expressions that help us break down complex expressions into simpler factors. One commonly used identity is the sum of cubes, which is useful in our problem here. It is expressed as:\(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\)This identity helps us factor cubic expressions conveniently, using a known pattern.
In the given task, we took\( x = \sqrt[3]{3} \) and\( y = 1 \).
Then, by substituting these values, we employ the identity to simplify the expression involving cubes.

• Identities make algebraic manipulation manageable by providing a structured format for factoring.
• They are essential tools for solving polynomial equations, simplifying, and rationalizing expressions.

Using these types of identities is particularly helpful when dealing with polynomials, because they can reduce complex expressions to simpler and more workable forms.

Algebraic Simplification

Algebraic simplification refers to the process of rewriting an algebraic expression in a simpler or more convenient form. In this particular exercise, we focus on simplifying the expression \( \frac{1}{\sqrt[3]{3}+1} \) by eliminating the irrational number in the denominator.
The core idea is to "rationalize" the denominator: to remove the cube root or any other irrational number.

• To simplify, we find a conjugate based on the expression: \( 3-\sqrt[3]{3}+1 \).
• Multiply both the numerator and the denominator by this conjugate to rationalize the expression (Step 5 of the solution).
• By utilizing algebraic properties, such as the factoring identity, the expression becomes rationalized.

This process involves multiplying out and simplifying both the numerator and denominator until the unwanted terms cancel out or become simplified further. Following these steps results in a neat fractional form \( \frac{3-\sqrt[3]{3}+1}{4} \), which has a rationalized denominator.Algebraic simplification is crucial for solving equations, performing integration, differentiation, and general calculations in mathematics.