Problem 66 - Algebra and Trigonometry

Question

Rationalize the denominator. (a) \(\frac{1}{\sqrt[3]{x^{2}}}\) (b) \(\frac{1}{\sqrt[4]{x^{3}}}\) (c) \(\frac{1}{\sqrt[3]{x^{4}}}\)

Step-by-Step Solution

Answer

(a) \(\frac{\sqrt[3]{x}}{x}\); (b) \(\frac{\sqrt[4]{x}}{x}\); (c) \(\frac{\sqrt[3]{x^{2}}}{x^{2}}\).

Step 1a: Identify the Denominator

For part (a), our task is to rationalize the denominator \( \sqrt[3]{x^{2}} \). This is a cube root expression and we aim to eliminate the radical from the denominator.

Step 2a: Multiply by a Conjugate

To rationalize \( \sqrt[3]{x^{2}} \), we multiply both the numerator and the denominator by \( \sqrt[3]{x} \). This makes the denominator a cube without a radical, because \( \sqrt[3]{x^{2}} \times \sqrt[3]{x} = x \).

Step 3a: Simplify the Expression

After the multiplication, the new expression becomes \( \frac{\sqrt[3]{x}}{x} \). Now, the denominator is rationalized.

Step 1b: Identify the Denominator

For part (b), we need to rationalize \( \sqrt[4]{x^{3}} \). This is a fourth root, and we need to remove this radical from the denominator.

Step 2b: Multiply by a Conjugate

To achieve this, multiply both the numerator and the denominator by \( \sqrt[4]{x} \). This will make the denominator \( \sqrt[4]{x^{3}} \times \sqrt[4]{x} = x \).

Step 3b: Simplify the Expression

After multiplying, the resulting expression is \( \frac{\sqrt[4]{x}}{x} \). Hence, the denominator is now rational.

Step 1c: Identify the Denominator

For part (c), we aim to rationalize \( \sqrt[3]{x^{4}} \). This is another cube root expression.

Step 2c: Multiply by a Conjugate

We multiply by \( \sqrt[3]{x^{2}} \) to make the denominator \( x^{2} \). So, we multiply \( \sqrt[3]{x^{4}} \times \sqrt[3]{x^{2}} = x^{2} \).

Step 3c: Simplify the Expression

The expression becomes \( \frac{\sqrt[3]{x^{2}}}{x^{2}} \), successfully rationalizing the denominator.

Key Concepts

Cube Root

The cube root of a number, symbolized as \( \sqrt[3]{x} \), is a value that, when multiplied by itself three times, yields the original number \( x \). Understanding cube roots is essential in algebra, especially when dealing with polynomial functions and simplifying expressions with radicals. In the context of rationalizing denominators, cube roots are quite common.
To rationalize a cube root expression like \( \frac{1}{\sqrt[3]{x^2}} \), we should eliminate the radical from the denominator. This involves multiplying both the numerator and the denominator by a term that can make the entire denominator a perfect cube. In this case, multiplying by \( \sqrt[3]{x} \) means that we adjust the power so that the radical becomes a simple exponent, like \( \sqrt[3]{x^2} \times \sqrt[3]{x} = x \).
By following these steps, you simplify the denominator, making your final algebraic expression easier to work with and understand. While it may seem complex initially, breaking down cube roots in this way often clarifies what can otherwise appear to be intricate algebraic expressions.

Fourth Root

The fourth root of a number, represented by \( \sqrt[4]{x} \), gives a value that, when multiplied by itself four times, results in the original number \( x \). Much like with square and cube roots, fourth roots are pivotal in higher-order algebraic simplifications.
When tasked with rationalizing a fourth root in the denominator such as \( \frac{1}{\sqrt[4]{x^3}} \), the aim is to remove the radical for simplification. To do this, you multiply both top and bottom of the fraction by \( \sqrt[4]{x} \). This process of multiplication ensures that the radical in the denominator becomes a whole exponent \( x \).
So, \( \sqrt[4]{x^3} \times \sqrt[4]{x} = x \), thereby eliminating the fourth root and leaving you with the expression \( \frac{\sqrt[4]{x}}{x} \). The process of rationalizing involves ensuring that the denominator is free of radicals, resulting in more easily interpretable algebraic expressions.

Algebraic Expression Simplification

Simplifying algebraic expressions involves reducing them to their simplest form without altering their value. This can include actions such as combining like terms, reducing fractions, and removing radicals.
For radical expressions like \( \frac{1}{\sqrt[3]{x^4}} \), simplification often focuses on removing radicals from the denominator. By doing so, expressions become simpler and appear tidier, which is especially helpful in solving complex equations or further algebraic manipulation.
In the example \( \frac{1}{\sqrt[3]{x^4}} \), we simplify by multiplying numerator and denominator by \( \sqrt[3]{x^2} \). This operation ensures the denominator becomes a power of \( x \): \( \sqrt[3]{x^4} \times \sqrt[3]{x^2} = x^2 \).
Thus, the expression \( \frac{\sqrt[3]{x^2}}{x^2} \) is simplified and devoid of radicals in its denominator. It's important to grasp these simplification techniques as they are fundamental in algebra, enabling cleaner solutions and better insights into the nature of algebraic functions.