Problem 3 - Algebra and Trigonometry

Question

Write each radical expression using exponents, and each exponential expression using radicals. Radical expression \(\quad\) Exponential expression _______ \(\quad\) \(4^{2 / 3}\)

Step-by-Step Solution

Answer

Exponential: \(4^{2/3}\) is \(\sqrt[3]{4^2}\) in radical form.

Step 1: Understanding Exponential Expressions

Exponential expressions like \(4^{2/3}\) can be rewritten in terms of radicals. Here, the exponent \(\frac{2}{3}\) indicates the power and the root: the number 4 is to be raised to the power of 2, and then the cube root is taken.

Step 2: Convert Exponential to Radical

Convert \(4^{2/3}\) to a radical form. The expression can be rewritten as \(\sqrt[3]{4^2}\). This means you take the cube root of 4 squared.

Step 3: Understanding Radical Expressions

Now, let's reverse the process. Consider a radical expression, such as \(\sqrt[n]{a^m}\). This indicates that you take the \(n\)-th root of \(a\), and then raise the result to the power of \(m\).

Step 4: Write Radical as Exponential

The expression \(\sqrt[3]{4^2}\) can be equivalently written as an exponent: \((4^2)^{1/3}\) or, more commonly, simply as \(4^{2/3}\), confirming the conversion is correct both ways.

Key Concepts

Radical Expressions

Radical expressions are mathematical expressions that involve roots. Most commonly, they deal with square roots, cube roots, or higher order roots. A radical expression typically looks like this: \( \sqrt[n]{a} \), where \( n \) is the degree of the root, and \( a \) is the number inside the radical sign. Here are some key points about radical expressions:

• The radical sign \( \sqrt{} \) indicates the root. If there is no number written, it is assumed to be a square root, or \( \sqrt[2]{} \).
• The number under the radical sign is the radicand. It's the value that you're taking the root of.
• The index, \( n \), represents which root you are dealing with (e.g., square root \( n=2 \), cube root \( n=3 \), etc.).

For example, in the expression \( \sqrt[3]{4^2} \), you take the third root (or cube root) of \( 4^2 \). This involves calculating \( 4 \) to the power of 2 first, and then taking the cube root of that result. This knowledge is essential when converting between radicals and exponents.

Exponential Expressions

Exponential expressions are expressions that involve exponents, which denote powers or repeated multiplication of a base number. An exponential expression looks like this: \( a^m \), where \( a \) is the base and \( m \) is the exponent. Here are a few important points about exponential expressions:

• The base \( a \) is the number that gets multiplied by itself as many times as indicated by the exponent.
• The exponent \( m \) tells you how many times to use the base as a factor in a multiplication.
• An exponent of \( 1 \) means the number is itself, and an exponent of \( 0 \) means the number is \( 1 \) (as long as the base is not zero).

The expression \( 4^{2/3} \) is an example of an exponential expression involving a fractional exponent. This means you perform two operations: raise 4 to the power of 2 and then take the cube root of the result. Understanding these expressions is crucial as they often need to be translated into radical form to simplify or solve.

Conversion Between Forms

Conversion between radicals and exponents is a powerful tool in mathematics that allows for easier manipulation of expressions. Here's how you convert between these forms:

• A radical expression \( \sqrt[n]{a^m} \) can be rewritten as an exponent: \( (a^m)^{1/n} \) or more concisely \( a^{m/n} \).
• An exponential expression with a fractional exponent \( a^{m/n} \) can be rewritten as a radical: \( \sqrt[n]{a^m} \).

This conversion is useful because it provides flexibility in solving equations and simplifying expressions. For example, converting \( 4^{2/3} \) into \( \sqrt[3]{4^2} \) makes it easier to calculate the cube root of the squared number, and it confirms the dual nature of radical and exponential expressions. Understanding how to switch between these forms expands the tools at your disposal when tackling mathematical problems.