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
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 $x^{\frac{\cancel{3} \cdot 1}{\cancel{3} \cdot 2}}$.
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.
