Problem 66 - Elementary and Intermediate Algebra

Question

Rationalize each denominator. $$ \frac{1}{\sqrt[5]{2}} $$

Step-by-Step Solution

Answer

The rationalized form is \( 2^{-1/5} \).

Step 1: Understand the Problem

We need to eliminate the radical from the denominator of the expression \( \frac{1}{\sqrt[5]{2}} \) by rationalizing it. This involves multiplying the numerator and the denominator by an appropriate expression to make the denominator a whole number.

Step 2: Find the Appropriate Factor

To rationalize \( \frac{1}{\sqrt[5]{2}} \), we need to multiply by a factor that will turn \( \sqrt[5]{2} \) into \( 2 \). Since \( (\sqrt[5]{2})^5 = 2 \), we should multiply by \( (\sqrt[5]{2})^4 \), the fourth power, to achieve this.

Step 3: Multiply Numerator and Denominator

Multiply both numerator and denominator by \( (\sqrt[5]{2})^4 \). This gives us:\[\frac{1}{\sqrt[5]{2}} \times \frac{(\sqrt[5]{2})^4}{(\sqrt[5]{2})^4} = \frac{(\sqrt[5]{2})^4}{2}\]

Step 4: Simplify the Expression

The expression \( (\sqrt[5]{2})^4 \) simplifies to \( 2^{4/5} \). Thus, the fraction becomes:\[\frac{2^{4/5}}{2}\]This expression can be further simplified by rewriting the numerator as powers of the denominator, resulting in:\[\frac{2^{4/5}}{2^1} = 2^{4/5 - 1} = 2^{-1/5}\]

Step 5: Final Result

Therefore, the expression \( \frac{1}{\sqrt[5]{2}} \) rationalized is \( 2^{-1/5} \).

Key Concepts

Rational Exponents

Rational exponents provide a convenient way to express roots as fractions in an exponent form. For example, the fifth root of a number can be written as an exponent with a denominator of five. This means we can write \( \sqrt[5]{2} \) as \( 2^{1/5} \). Rational exponents follow the same rules as integer exponents, making it easier to manipulate expressions involving roots. They allow us to rewrite expressions like \( (\sqrt[5]{2})^4 \) as \( 2^{4/5} \). This conversion helps simplify algebraic expressions and is useful in rationalizing denominators.

Algebraic Expressions

Algebraic expressions are combinations of numbers and variables connected by operations such as addition, subtraction, multiplication, and division. These expressions, such as \( \frac{1}{\sqrt[5]{2}} \), are used extensively in mathematics to represent relationships and solve equations. When simplifying algebraic expressions, the goal is often to write the expression in its simplest form, which may involve rewriting radicals or rational exponents. Rationalizing the denominator is a common technique used to transform algebraic expressions into a more standard form where the denominator is a rational number.

Simplifying Radicals

Simplifying radicals involves rewriting them so they are easier to work with. In the expression \( \frac{1}{\sqrt[5]{2}} \), we aim to remove the radical from the denominator, which involves multiplying by a strategically chosen expression. In the provided solution, we multiply by the fourth power of the radical, \( (\sqrt[5]{2})^4 \), because raising \( \sqrt[5]{2} \) to the power of five produces a whole number. By simplifying \( (\sqrt[5]{2})^4 \) to \( 2^{4/5} \), the fraction \( \frac{2^{4/5}}{2} \) is obtained. Converting radicals to exponents with fractions allows us to apply properties of exponents to simplify further, turning \( 2^{4/5 - 1} \) into \( 2^{-1/5} \). This step-by-step approach ultimately leads to a simpler, more manageable form.