Problem 60 - Algebra 2 and Trigonometry

Question

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ 6^{\frac{1}{3}} $$

Step-by-Step Solution

Answer

The radical expression is \(\sqrt[3]{6}\).

Step 1: Understand the Expression

The expression given is \(6^{\frac{1}{3}}\). In this expression, the exponent \(\frac{1}{3}\) indicates that we need to find the cube root of 6. The expression needs to be rewritten in radical form.

Step 2: Convert the Power to a Radical

The general form to convert a fractional exponent to a radical is \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\). For \(6^{\frac{1}{3}}\), \(m\) is 1 and \(n\) is 3. Hence it converts to \(\sqrt[3]{6^1}\).

Step 3: Simplify the Radical Expression

Since \(6^1 = 6\), the expression \(\sqrt[3]{6^1}\) simplifies to \(\sqrt[3]{6}\). This is the simplest form of the expression as a radical.

Key Concepts

Fractional Exponents

Fractional exponents, like in the expression \(6^{\frac{1}{3}}\), are a way of expressing roots using powers. The numerator of the fraction represents the power, and the denominator indicates the root. For example, in \(a^{\frac{m}{n}}\):

• \(m\) is the power, showing how many times to multiply the base \(a\).
• \(n\) is the root, ascertaining which root to take of the base.

Understanding fractional exponents is crucial because they bridge the concepts of powers and roots. They allow for a more flexible notation and often make calculations easier to handle, especially with complex numbers. In more tangible terms, \(6^{\frac{1}{3}}\) means we are finding the number which, when cubed, gives us 6, which is expressed as the cube root of 6. This is a foundational concept in algebra and is particularly useful in calculus and higher mathematics.

Radical Form Conversion

Converting a power expressed with a fractional exponent to a radical form is a skill useful in simplifying expressions. To convert \(a^{\frac{m}{n}}\) into its equivalent radical form, follow this structure: \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\). Here, \(\sqrt[n]{ }\) indicates the \(n\)-th root, while \(a^m\) suggests the base \(a\) raised to the power of \(m\). Let's look at \(6^{\frac{1}{3}}\). Applying the conversion rule, we get \(\sqrt[3]{6^1}\). Broken down:

• \(m = 1\), so we raise 6 to the first power, which remains 6.
• \(n = 3\), indicating the cube root.

The process simplifies the expression without altering its value. This conversion is particularly handy because it translates the expression into a form that can be more readily evaluated or simplified further if possible.

Simplifying Radicals

Simplifying radicals involves expressing them in their most reduced form. This process makes calculations easier and expressions neater. Consider \(\sqrt[3]{6}\), which is already simplified as it's expressed directly in terms of its prime factorization. Here's why:

• There are no factors of 6 that can be expressed as perfect cubes, hence no further simplification is possible.

When simplifying any radical, check if the radicand (in this case, 6) can be broken into smaller parts that are cubes, squares, or any other power corresponding to your root (cube root in this instance). Since 6 is composed of 2 and 3, neither of which is a cube, \(\sqrt[3]{6}\) is already as simple as it gets. Working with radicals often involves expressing them in a way that reveals insights about their characteristics or values, making simplification a key part of algebraic manipulation.