To determine which expression is equivalent to [tex]\(x^{\frac{1}{6}} \cdot x^{\frac{1}{6}}\)[/tex], we can follow these steps:
1. Understand the properties of exponents: When multiplying two expressions with the same base, we add their exponents. This is given by:
[tex]\[ x^a \cdot x^b = x^{a + b} \][/tex]
2. Identify the exponents in the given expression: Here, we have [tex]\( x^{\frac{1}{6}} \cdot x^{\frac{1}{6}} \)[/tex]. Both exponents are [tex]\(\frac{1}{6}\)[/tex].
3. Add the exponents: According to the property mentioned in step 1, we add the exponents:
[tex]\[ \frac{1}{6} + \frac{1}{6} = \frac{2}{6} \][/tex]
4. Simplify the exponent: The fraction [tex]\(\frac{2}{6}\)[/tex] can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2:
[tex]\[ \frac{2}{6} = \frac{2 \div 2}{6 \div 2} = \frac{1}{3} \][/tex]
5. Rewrite the expression: After simplifying, the expression becomes:
[tex]\[ x^{\frac{2}{6}} = x^{\frac{1}{3}} \][/tex]
6. Identify the equivalent expression: The expression [tex]\( x^{\frac{1}{3}} \)[/tex] is equivalent to the cube root of [tex]\(x\)[/tex]. This can be written as:
[tex]\[ \sqrt[3]{x} \][/tex]
Therefore, the expression equivalent to [tex]\( x^{\frac{1}{6}} \cdot x^{\frac{1}{6}} \)[/tex] is:
[tex]\[ \sqrt[3]{x} \][/tex]
So, the correct answer is:
[tex]\[ \sqrt[3]{x} \][/tex]
