Problem 7 - Algebra and Trigonometry

Question

Write each radical expression using exponents, and each exponential expression using radicals. Radical expression \(\quad\) Exponential expression _______ \(\quad\) \(a^{2 / 5}\)

Step-by-Step Solution

Answer

The radical expression is \( \sqrt[5]{a^2} \).

Step 1: Understanding Radical and Exponential Notation

A radical expression is generally of the form \( \sqrt[n]{a^m} \) which can be translated into an exponential expression as \( a^{m/n} \). Here, the exponent \( m \) is the power and \( n \) is the root. Similarly, an exponential expression \( a^{m/n} \) can be converted back into a radical as \( \sqrt[n]{a^m} \).

Step 2: Convert Exponential Expression to Radical

Given the exponential expression \( a^{2/5} \), we need to express it as a radical. The expression \( a^{2/5} \) indicates that the base \( a \) is raised to the power of 2, and then the fifth root is taken. Therefore, \( a^{2/5} \) can be written in radical form as \( \sqrt[5]{a^2} \).

Step 3: Write the Radical Expression

From Step 2, we determined that the radical corresponding to the exponential expression \( a^{2/5} \) is \( \sqrt[5]{a^2} \). This means that for the exponential expression \( a^{2/5} \), the equivalent radical expression is \( \sqrt[5]{a^2} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, where a number or variable is found under a radical symbol, typically shown as \( \sqrt{} \). The radical itself represents a root, such as a square root, cube root, or any other specified root. Radicals include an index and a radicand. - **Index**: This small number placed before the radical sign tells us which root is considered. If no index is displayed, it is assumed to be a square root (index 2). For example, \( \sqrt{a} \) is the square root of \( a \), and \( \sqrt[3]{a} \) is the cube root.- **Radicand**: The number or expression found under the radical sign.Converting between radical and exponential forms can simplify complex mathematical operations.

Exponents

Exponents are used to express powers of numbers or variables. An exponential expression is written with a base and an exponent, such as \( a^b \), where \( a \) is the base and \( b \) is the exponent.- **Base**: This is the number or variable that is being multiplied.- **Exponent**: This tells us how many times the base is multiplied by itself. For instance, \( a^3 \) means \( a \times a \times a \).Exponents simplify the representation of repeated multiplication, making calculations more straightforward.

Radicals

Radicals refer to expressions involving roots, utilizing the radical symbol (\( \sqrt{} \)) to denote which root to take of a certain number.- **Structure of a Radical**: Radicals are often represented in the form \( \sqrt[n]{a} \), where \( n \) is the index and \( a \) is the radicand. The index \( n \) indicates the degree of the root. If not explicitly mentioned, it's assumed to be 2, implying a square root.Radicals provide a way to express roots that might otherwise be complex to handle, especially in algebra where we frequently simplify or manipulate such expressions.

Convert Exponential to Radical

Converting an exponential expression to a radical involves transforming it from a form like \( a^{m/n} \) into \( \sqrt[n]{a^m} \). This change helps visualize the operation as a combination of both power and root.Steps to Convert:

• Identify the base \( a \), the exponent numerator \( m \), and the denominator \( n \) from the exponential expression \( a^{m/n} \).
• Rewrite this as a radical \( \sqrt[n]{a^m} \) where \( n \) becomes the radical's index, and \( a^m \) becomes the radicand raised to the power \( m \).

For instance, given \( a^{2/5} \), the conversion results in \( \sqrt[5]{a^2} \). This shows \( a \) raised to the power of 2, while taking the fifth root, blending both operations into one approachable expression.