Problem 89 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 85-90: Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. $$ \sqrt[n]{\frac{1}{c}} \square \frac{1}{\sqrt[n]{c}} $$

Step-by-Step Solution

Answer

Replace \( \square \) with \( = \).

Step 1: Simplify the Left Side

The left side of the equation is \( \sqrt[n]{\frac{1}{c}} \). This expression represents the \( n \)-th root of \( \frac{1}{c} \). Using the property of exponents for roots, this can be rewritten as \( \left( \frac{1}{c} \right)^{1/n} \).

Step 2: Simplify the Right Side

The right side of the equation is \( \frac{1}{\sqrt[n]{c}} \). This expression represents the reciprocal of the \( n \)-th root of \( c \). Again using properties of roots, it can be rewritten as \( c^{-1/n} \).

Step 3: Compare the Expressions

Now compare \( \left( \frac{1}{c} \right)^{1/n} \) and \( c^{-1/n} \). Rewrite \( \left( \frac{1}{c} \right)^{1/n} \) as \( c^{-1/n} \) since \( \frac{1}{c} = c^{-1} \). Thus, \( c^{-1/n} = c^{-1/n} \). The two expressions are equivalent.

Step 4: Determine the Appropriate Symbol

Since both expressions \( \sqrt[n]{\frac{1}{c}} \) and \( \frac{1}{\sqrt[n]{c}} \) simplify to the same expression \( c^{-1/n} \), they are equal. We replace \( \square \) with \( = \) to make the statement true.

Key Concepts

Properties of Exponents

Exponents are powerful tools in mathematics that help in simplifying expressions and solving complex equations with ease. One important property of exponents is how they deal with roots:

• The expression \( a^{1/n} \) represents the \( n \)-th root of \( a \). Therefore, if you have \( \sqrt[n]{a} \), it is equivalent to \( a^{1/n} \).
• When dividing exponents, we apply the rule: \( a^{m}/a^{n} = a^{m-n} \).

These, along with the reciprocal property, enable us to rewrite expressions in a simple manner. For example, in our exercise, we used these properties to rewrite \( \sqrt[n]{\frac{1}{c}} \) as \( \left(\frac{1}{c}\right)^{1/n} \), which further simplifies to \( c^{-1/n} \) using exponents. This clear translates operations between roots and exponents to simpler form, highlighting their utility in mathematics.

Nth Root Simplification

Simplifying expressions involving the n-th root can often seem daunting at first. However, by expressing roots as exponents, simplification becomes more straightforward.

• Consider \( \sqrt[n]{a} \), which translates to \( a^{1/n} \). This transformation allows us to use all the rules of exponents to simplify or manipulate expressions.
• In fractional representations like \( \sqrt[n]{\frac{1}{c}} \), you can distribute the exponent: \( \left(\frac{1}{c}\right)^{1/n} = c^{-1/n} \).

This shows how exponents provide a clear and systematic method for simplifying radical expressions. By breaking them down into smaller, more manageable steps, you can tackle even complex n-th root expressions with confidence.

Reciprocals

Reciprocals are fundamental in dividing fractions and handling roots. The reciprocal of a number \( a \) is \( \frac{1}{a} \), and this concept extends to roots and exponents seamlessly.

• For expressions like \( \sqrt[n]{\frac{1}{c}} \), understanding that it can be rewritten as the reciprocal of \( \sqrt[n]{c} \), which is \( \frac{1}{\sqrt[n]{c}} \), assists in simplifying expressions.
• In exponent terms, the reciprocal translates to \( c^{-1/n} \) using properties like \( a^{-n} = \frac{1}{a^{n}} \).

By leveraging reciprocals, you can convert complex expressions into forms that are easier to manage, facilitating comparison, simplification, and further computation.

Equality and Inequality Symbols

Symbols like \( = \) and \( eq \) play a crucial role in expressing mathematical statements. They tell us if two expressions are equal or not, representing a key part of solving equations or validating statements.

• Using properties, we simplified both sides of our given equation to show \( c^{-1/n} = c^{-1/n} \). Hence, the equality symbol \( = \) is used.
• Understanding when to use inequality symbols \( eq \) requires careful attention to detail, ensuring you properly simplify and compare expressions.

These symbols are essential for establishing relationships between numbers or expressions, allowing us to communicate mathematical truths effectively. They are foundational in all levels of mathematics, from basic arithmetic to advanced algebra.