Question

Using the properties of exponents, which expression is equivalent to x to the square root 1/6 ⋅x 1/6 ?(1 point) Responses 1x√3 Start Fraction 1 over root index 3 Start Root x End Root End Fraction x−−√ Start Root x End Root x−−√3 root index 3 Start Root x End Root x−−√36

Answer 1

To simplify the expression \( x^{\sqrt{\frac{1}{6}}} \cdot x^{\frac{1}{6}} \), you can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \).

1. The first term is \( x^{\sqrt{\frac{1}{6}}} \).
2. The second term is \( x^{\frac{1}{6}} \).

Now sum the exponents:

\[
\sqrt{\frac{1}{6}} + \frac{1}{6}
\]

To add these two fractions, it's helpful to express \(\sqrt{\frac{1}{6}}\) in terms of a fraction. We know that:

\[
\sqrt{\frac{1}{6}} = \frac{1}{\sqrt{6}}
\]

Next, rationalize the denominator:

\[
\frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{6}}{6}
\]

Now we add \(\frac{\sqrt{6}}{6}\) and \(\frac{1}{6}\):

\[
\frac{\sqrt{6} + 1}{6}
\]

This means:

\[
x^{\sqrt{\frac{1}{6}}} \cdot x^{\frac{1}{6}} = x^{\frac{\sqrt{6} + 1}{6}}
\]

None of the provided responses seem to represent this expression directly. Could you please clarify what format or simplification is expected for the answer or recheck the response options?