Problem 88 - Algebra and Trigonometry

Question

\(83-88=\) Rationalize the denominator. $$ \begin{array}{llll}{\text { (a) } \frac{1}{\sqrt[3]{x^{2}}}} & {\text { (b) } \frac{1}{\sqrt[4]{x^{3}}}} & {\text { (c) } \frac{1}{\sqrt[3]{x^{4}}}}\end{array} $$

Step-by-Step Solution

Answer

Rationalized forms: (a) \(\frac{\sqrt[3]{x}}{x}\), (b) \(\frac{\sqrt[4]{x}}{x}\), (c) \(\frac{\sqrt[3]{x^{2}}}{x^{2}}\).

Step 1: Understand the Problem

We need to rationalize the denominators for each of the given expressions. This involves removing any roots present in the denominator by multiplying by a suitable expression.

Step 2: Rationalize Part (a)

For expression (a) \(\frac{1}{\sqrt[3]{x^{2}}}\), we need to multiply by a form of 1 that will allow the exponent on \(x\) inside the radical to become 3. The multiplier is \(\frac{\sqrt[3]{x}}{\sqrt[3]{x}}\). This results in \(\frac{\sqrt[3]{x}}{x}\).

Step 3: Rationalize Part (b)

For expression (b) \(\frac{1}{\sqrt[4]{x^{3}}}\), to eliminate the radical, multiply by \(\frac{\sqrt[4]{x}}{\sqrt[4]{x}}\). This turns the expression into \(\frac{\sqrt[4]{x}}{x}\).

Step 4: Rationalize Part (c)

For expression (c) \(\frac{1}{\sqrt[3]{x^{4}}}\), multiply by \(\frac{\sqrt[3]{x^{2}}}{\sqrt[3]{x^{2}}}\). This results in \(\frac{\sqrt[3]{x^{2}}}{x^{2}}\), effectively removing the cube root from the denominator.

Key Concepts

Cube Root

A cube root is a special number which, when multiplied by itself three times, gives the original number. In mathematical notation, the cube root of a number \(x^2\) is represented as \( \sqrt[3]{x^2} \). Understanding how cube roots work is crucial when simplifying algebraic expressions. When dealing with an expression like \( \frac{1}{\sqrt[3]{x^2}} \), you need to learn how to rationalize the denominator. Rationalization is the process of eliminating the root from the denominator. You multiply both the numerator and the denominator by a term that will make the exponent inside the radical a perfect cube. For example:

• Here, the square on \(x\) means you need one more factor of \(x\) to get a perfect cube. So, multiply by \( \frac{\sqrt[3]{x}}{\sqrt[3]{x}} \) to achieve a denominator of \( x \).
• This turns the expression into \( \frac{\sqrt[3]{x}}{x} \).

By performing these steps, you eliminate the cube root from the denominator, simplifying the expression.

Fourth Root

A fourth root is similar in concept to a cube root, but with the number being multiplied four times to obtain the original value. It is symbolized as \( \sqrt[4]{x^3} \). Understanding fourth roots is particularly useful for simplifying expressions where you must remove this type of root from the denominator. Take an expression like \( \frac{1}{\sqrt[4]{x^3}} \). The goal is to rationalize the denominator to avoid a radical:

• Since you have \(x^3\) in the radical, you'll need one more factor of \(x\) to turn it into a perfect fourth power. Hence, multiply by \( \frac{\sqrt[4]{x}}{\sqrt[4]{x}} \).
• This results in the simplified form \( \frac{\sqrt[4]{x}}{x} \).

This process removes the fourth root from the denominator, leaving a simpler arithmetic fraction.

Algebraic Fractions

Algebraic fractions are expressions formed by dividing one polynomial by another, where the denominator only has variables. Rationalizing the denominator of such fractions involves skillfully manipulating the expression so that no roots appear in the denominator. For instance, if dealing with \( \frac{1}{\sqrt[3]{x^4}} \), you need to:

• Multiply by \( \frac{\sqrt[3]{x^2}}{\sqrt[3]{x^2}} \). This choice is strategic to make the denominator a cube, specifically \( x^2 \), thereby removing the radical.
• The final form of the expression is \( \frac{\sqrt[3]{x^2}}{x^2} \).

By carefully selecting the multiplier based on the original power, the algebraic fraction becomes easier to interpret and use.

Simplifying Radicals

Simplifying radicals involves reducing the expression under the root sign into the simplest possible form. This is often done to make mathematical manipulation easier, especially when the radical appears in the denominator. Key steps when simplifying radicals include:

• Look at the radicand (the number under the root) and determine if it can be separated into simpler components that are perfect powers of the root.
• Factor out these components and simplify the expression by performing root calculations where applicable.
• Apply similar steps for rationalizing denominators where the expression is turned into an equivalent simpler form without radicals in the denominator.

By mastering simplification, numbers become less complex and the calculations are generally cleaner, allowing for a deeper understanding of the problem at hand.