Convert to Radical Form x^(3/6)

The question is asking for the conversion of an expression with an exponent written as a fraction into its equivalent radical form. Specifically, the expression given is x raised to the power of three-sixths (3/6), and the task is to rewrite this expression using a radical sign (also known as a root sign) to represent the fractional exponent in a different mathematical notation. Essentially, the question is about applying the understanding of the relationship between exponents and radicals to express the given power in terms of a root.

$x^{\frac{3}{6}}$

Answer

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Solution:

Step 1: Simplify the Exponent

Reduce the fraction $\frac{3}{6}$ to its simplest form by eliminating common factors.

Step 1.1: Identify Common Factors

Separate the numerator 3 as $3 \cdot 1$ . Thus, we have $x^{\frac{3 \cdot 1}{6}}$ .

Step 1.2: Factor Out the Common Factor

Recognize that 3 is a common factor in both the numerator and denominator. We express it as $x^{\frac{3 \cdot 1}{3 \cdot 2}}$ .

Step 1.3: Simplify the Fraction

Eliminate the common factor of 3 from both the numerator and denominator, resulting in $x^{\frac{3 \cdot 1}{3 \cdot 2}}$ .

Step 1.4: Finalize the Simplified Exponent

The expression simplifies to $x^{\frac{1}{2}}$ .

Step 2: Convert Exponent to Radical Form

Use the property $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to transform the exponent into a radical, which gives us $\sqrt{x^{1}}$ .

Step 3: Simplify the Radical Expression

Recognize that any number raised to the power of 1 is the number itself, so we have $\sqrt{x}$ .

Knowledge Notes:

To convert an expression from exponential form to radical form, you need to understand the following concepts:

Fraction Simplification: Fractions can be simplified by dividing both the numerator and the denominator by their greatest common factor (GCF). In this case, the GCF of 3 and 6 is 3.
Exponent Rules: The rule for converting an exponent to a radical is based on the property $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ , where $m$ is the power and $n$ is the root.
Radicals: A radical expression $\sqrt[n]{x}$ represents the $n$ -th root of $x$ . When $n$ is 2, it is commonly referred to as the square root and the index is often omitted.
Identity Property of Exponentiation: Raising any base to the power of 1 results in the base itself, i.e., $x^{1} = x$ .
LaTeX Formatting: To properly display mathematical expressions in a readable format, LaTeX is used. For instance, $\frac{3}{6}$ is rendered as a fraction, and $\sqrt{x}$ is rendered as a square root.

Understanding these concepts is crucial for correctly converting expressions between exponential and radical forms.