Problem 12 - Elementary and Intermediate Algebra

Question

Complete each rule for exponents. $$ x^{m / n}=\square=\sqrt[n]{x^{m}} $$

Step-by-Step Solution

Answer

\(x^{m/n} = (\sqrt[n]{x})^{m} = \sqrt[n]{x^m}\).

Step 1: Understanding the Exponentiation Rule

The expression \(x^{m/n}\) denotes raising \(x\) to the power of a fraction \(\frac{m}{n}\). The numerator \(m\) and the denominator \(n\) have distinct roles in the equation. Let's explore what these roles are.

Step 2: Using Fractional Exponents

A fractional exponent \(\frac{m}{n}\) means two things: the denominator \(n\) represents what root of the base \(x\) we are taking, and the numerator \(m\) represents the power to which the result should be raised. Hence, we understand \(x^{m/n}\) in terms of roots and powers.

Step 3: Converting to Radicals

To convert \(x^{m/n}\) into a radical expression, recognize that \(x^{m/n} = \sqrt[n]{x^m}\). This expression signifies taking the \(n\)-th root of \(x\) raised to the power of \(m\).

Step 4: Filling in the Missing Part

Given the equation \(x^{m/n} = \square = \sqrt[n]{x^{m}}\), we can fill in the box as follows: \(x^{m/n} = (\sqrt[n]{x})^{m}\). This illustrates that the \(n\)-th root of \(x\) is taken first, and then it is raised to the power of \(m\).

Key Concepts

Exponentiation Rules

Exponentiation rules provide a framework for dealing with powers in mathematics. These rules are essential for simplifying expressions and performing arithmetic operations that involve powers. When dealing with fractional exponents like \(x^{m/n}\), it's important to understand that this expression combines two operations:

• First, \(n\) represents the type of root or radical being taken.
• Second, \(m\) indicates to which power the resulting value should be raised.

Understanding these roles helps simplify complex expressions and solve problems more efficiently. Applying these rules allows one to manipulate and transform expressions by changing between radical and exponent forms.
For example, knowing that \((x^a)^b = x^{ab}\) helps transition between different mathematical forms, making calculations easier.

Radical Expressions

Radical expressions are mathematical expressions that involve roots, such as square roots, cube roots, and so on. These are often written using the radical symbol \(\sqrt{}\). For fractional exponents, radical expressions express the operation of taking the root of a number. For instance, \(\sqrt[n]{x}\) denotes the \(n\)-th root of \(x\).

Converting a fractional exponent into a radical expression is straightforward. Using the rule \(x^{m/n} = \sqrt[n]{x^m}\), we see that radical expressions offer a different perspective for understanding powers and roots. This transformation is crucial when you need to integrate or differentiate expressions in calculus, as it often simplifies the process.

• Radicals offer another way to consider powers, making certain mathematical manipulations more convenient.
• They provide a visual means to understand the relationship between multiplication and roots.

Radical expressions often serve to simplify calculations, helping to visualize complex relationships in algebra.

Roots and Powers

Roots and powers are fundamental concepts in mathematics that relate to exponentiation and radicals. In understanding \(x^{m/n}\), these two concepts are intertwined:

• The root represented by \(n\) tells us how many times a number must be multiplied by itself to achieve the original number \(x\).
• The power \(m\) suggests that the result of the root operation is then taken to this power.

For example, in \(x^{1/2}\), the "1" indicates the square root, and the equivalent radical form would be \(\sqrt{x}\). When we deal with complex numbers or higher-level mathematics, understanding the translation between roots and powers using fractional exponents is crucial.

They simplify terms and allow us to solve and transform equations with ease. Conceptually, linking roots to denominators and powers to numerators in fractional exponents bridges the arithmetic and algebraic practices in mathematics, facilitating better comprehension and application.