Certainly! Let's solve the problem step-by-step using the properties of exponents.
We are given the expression [tex]\( x^{\frac{1}{6}} \cdot x^{\frac{1}{6}} \)[/tex].
1. Identify the property of exponents:
When multiplying two exponential expressions with the same base, we add their exponents. Mathematically, this is expressed as:
[tex]\[
a^m \cdot a^n = a^{m+n}
\][/tex]
where [tex]\( a \)[/tex] is the base and [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are the exponents.
2. Apply the property to the given expression:
In the given expression [tex]\( x^{\frac{1}{6}} \cdot x^{\frac{1}{6}} \)[/tex], the base is [tex]\( x \)[/tex] and the exponents are both [tex]\( \frac{1}{6} \)[/tex].
According to the property of exponents, we add the exponents together:
[tex]\[
x^{\frac{1}{6}} \cdot x^{\frac{1}{6}} = x^{\left(\frac{1}{6} + \frac{1}{6}\right)}
\][/tex]
3. Perform the addition of the exponents:
[tex]\[
\frac{1}{6} + \frac{1}{6} = \frac{1 + 1}{6} = \frac{2}{6} = \frac{1}{3}
\][/tex]
4. Rewrite the expression with the simplified exponent:
[tex]\[
x^{\left(\frac{1}{6} + \frac{1}{6}\right)} = x^{\frac{1}{3}}
\][/tex]
Thus, the expression [tex]\( x^{\frac{1}{6}} \cdot x^{\frac{1}{6}} \)[/tex] is equivalent to [tex]\( x^{\frac{1}{3}} \)[/tex].
So, the equivalent expression is [tex]\( x^{\frac{1}{3}} \)[/tex].
