Problem 65 - Elementary and Intermediate Algebra

Question

Rationalize each denominator. $$ \frac{1}{\sqrt[4]{8}} $$

Step-by-Step Solution

Answer

The rationalized form of \( \frac{1}{\sqrt[4]{8}} \) is \( 8^{-1/4} \) or \( \sqrt[4]{8^3} \).

Step 1: Express the Denominator Using Exponents

First, we need to express the fourth root as an exponent. The fourth root of 8 can be written with a fractional exponent: \( \sqrt[4]{8} = 8^{1/4} \). So, the expression becomes \( \frac{1}{8^{1/4}} \).

Step 2: Multiply by a Conjugate to Eliminate the Root

To rationalize the denominator, we need to eliminate the \( 8^{1/4} \) by finding a factor that will make the exponent in the power of 8 an integer. We can do this by multiplying the numerator and the denominator by \( 8^{3/4} \) because \( \frac{1}{4} + \frac{3}{4} = 1 \). Hence, we multiply by \( \frac{8^{3/4}}{8^{3/4}} \).

Step 3: Simplify the Expression

Multiply the numerator and the denominator by \( 8^{3/4} \). The expression becomes: \[ \frac{1 \cdot 8^{3/4}}{8^{1/4} \cdot 8^{3/4}} = \frac{8^{3/4}}{8^{1}} \].

Step 4: Re-write and Simplify the Fraction

Since \( 8^{1} = 8 \), the expression simplifies to \( \frac{8^{3/4}}{8} \). When dividing exponentials with the same base, subtract the exponents: \( 8^{3/4} \div 8 = 8^{3/4 - 1} = 8^{-1/4} \). Use this to write \( 8^{-1/4} \) as \( \frac{1}{8^{1/4}} \), which is the reciprocal in simplified radical form: \( \sqrt[4]{8} \).

Step 5: Final Simplified Rationalized Expression

The rationalized expression simplifies further to be \( 8^{-1/4} \), which is equal to \( \frac{1}{\sqrt[4]{8}} \), showing \( 8^{-1/4} \) as a rationalized form of the expression. But more traditionally shown, the rationalized form is \( 8^{1/4} \cdot 8^{-1} = \sqrt[4]{8^3} \).

Key Concepts

Fractional Exponents

Fractional exponents are a way of expressing roots as powers. When you see an expression like \( a^{m/n} \), it means the \( n \)-th root of \( a \) raised to the power of \( m \). For example, \( 8^{1/4} \) is the fourth root of 8.

• The denominator \( n \) in the fraction represents which root to take.
• The numerator \( m \) indicates to what power the root is raised.

This allows us to use the properties of exponents to simplify and manipulate expressions more easily. Transitioning from radical form \( \sqrt[4]{8} \) to \( 8^{1/4} \) helps us integrate it smoothly into algebraic operations.

Simplifying Exponents

Simplifying exponents involves using the rules of exponents to combine terms into a simpler form. In our example, by multiplying the denominator \( 8^{1/4} \) by \( 8^{3/4} \), we get an exponent of 1: \( 8^{1/4 + 3/4} = 8^{1} \).

• Add exponents together when multiplying like bases.
• Subtract them when dividing like bases.

Ultimately, when we have expressions like \( 8^{3/4} \div 8 \), we simplify by subtracting exponents to get \( 8^{-1/4} \). This application of operations with exponents, especially fractional ones, is central to simplifying and rationalizing expressions with roots.

Rational Expression

A rational expression is a fraction where the numerator and/or the denominator involves algebraic expressions. When dealing with rationalizing denominators, our goal is to convert the denominator into a rational number.
In the exercise, \( \frac{1}{8^{1/4}} \) is simplified by multiplying the numerator and denominator by \( 8^{3/4} \) to rationalize the denominator, transforming it into \( \frac{8^{3/4}}{8} \).

• Keep expressions equivalent by multiplying by forms of 1, like \( \frac{8^{3/4}}{8^{3/4}} \).
• This process ensures the expression’s value remains unchanged.

The rationalization process aims to remove roots from the denominator, making it easier to handle algebraically.

Algebraic Manipulation

Algebraic manipulation is key in reshaping expressions for simplification or evaluation. It involves operations such as multiplication, division, addition, or subtraction to achieve a desired form or factoring.
In this context, we're applying a clever form of multiplication to simplify an expression. By multiplying \( \frac{1}{8^{1/4}} \) by \( \frac{8^{3/4}}{8^{3/4}} \), we’re altering the form to simplify it, effectively clearing the root from the denominator.

• This technique highlights the understanding and utilization of mathematical equality.
• Algebraic manipulation leverages properties of numbers and operations to streamline expressions.

By mastering these manipulations, you gain control over the structure and form of mathematical expressions, making them easier to work with in more complex problems.