Problem 9 - Algebra and Trigonometry

Question

Write each radical expression using exponents, and each exponential expression using radicals. $$ 4^{2 / 3} $$

Step-by-Step Solution

Answer

\(4^{2/3} = \sqrt[3]{4^2}\)

Step 1: Understand the Problem

The task is to convert the given exponential expression, \(4^{2/3}\), into a radical expression. The exponent is a fraction, indicating a root and a power.

Step 2: Recall the Rule for Fractional Exponents

A fractional exponent \(a^{m/n}\) can be rewritten using radicals as \(\sqrt[n]{a^m}\). Here, \(m\) is the power and \(n\) is the root.

Step 3: Apply the Rule

For \(4^{2/3}\), we use the rule. The denominator 3 indicates a cube root and the numerator 2 indicates squaring the number. Thus, it can be rewritten as \((\sqrt[3]{4})^2\).

Step 4: Write the Radical Expression

The expression \((\sqrt[3]{4})^2\) can also be rewritten in a form that first involves taking the cube root and then squaring the result: \((\sqrt[3]{4})^2 = \sqrt[3]{4^2}\).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, or higher-order roots. When you see a radical symbol, it represents some form of a root. For example, \(\sqrt{9}\) represents the square root of nine, which is 3.
Understanding how to manipulate these expressions is important. You can transform radical expressions into exponential forms to simplify calculations or to handle algebraic operations.
Consider the expression \(\sqrt[n]{a^m}\). This is a general form of a radical expression that includes a rational exponent. Here, \(n\) denotes the order of the root, and \(m\) indicates the power of the radical's base. Recognizing these components allows you to convert between radical and exponential forms effectively.
For instance, to express \(\sqrt[3]{4^2}\) in its original exponential form, you would write \(4^{2/3}\). This conversion is key in solving many algebraic problems, where a simpler form of the expression can offer new insights or solutions.

Exponential Expressions

Exponential expressions involve numbers raised to a power. The expression consists of a base and an exponent. For instance, \(4^3\) is an exponential expression where 4 is the base, and 3 is the exponent, indicating 4 multiplied by itself three times.
Fractional exponents are a fascinating subset of exponential expressions. They combine both the power and root in one expression. When you see an expression like \(4^{2/3}\), it means you are dealing with both a root and a power. Specifically, it combines the cube root and the squared power.
Recognizing how to handle these exponents is critical. Understanding the relationship between roots and fractional exponents can simplify complex arithmetic and algebraic tasks. By converting an exponential expression to a radical form or vice versa, students can often make problems more manageable or solve them more intuitively.

Cube Root

The cube root is a specific type of radical expression used to find a number that, when multiplied by itself three times, equals the original number. For example, \(\sqrt[3]{8} = 2\) because \(2 \times 2 \times 2 = 8\).
Cube roots are especially common in problems involving volume because the cube's volume formula inherently includes a third power. In mathematics and physics, understanding cube roots can enable you to approach 3D problems with greater ease.
When dealing with fractional exponents where the denominator is 3, it's crucial to remember that you're working with a cube root. In the expression \(4^{2/3}\), the denominator (3) instructs us to take the cube root, while the numerator (2) tells us to square the result afterward.
By becoming comfortable with cube roots and their conversions between radical and exponential forms, students can enhance their problem-solving abilities and tackle more complex mathematical challenges.