Question

Using the properties of exponents, which expression is equivalent to x^1/6⋅x^1/6 ?(1 point)
Responses

x−−√3
root index 3 Start Root x End Root

1x√3
Start Fraction 1 over root index 3 Start Root x End Root End Fraction

3−−√x
Start Root x End Root

36−−√x

Answer 1

To simplify the expression \( x^{1/6} \cdot x^{1/6} \), we can use the property of exponents which states that when you multiply two powers with the same base, you add their exponents:

\[
x^{a} \cdot x^{b} = x^{a + b}
\]

In this case, we have:

\[
x^{1/6} \cdot x^{1/6} = x^{1/6 + 1/6} = x^{2/6} = x^{1/3}
\]

Now, we need to express \( x^{1/3} \) in terms of roots:

\[
x^{1/3} = \sqrt[3]{x}
\]

Now, we can match this with the provided options:

- \( \sqrt[3]{x} \)
- \( \frac{1}{\sqrt[3]{x}} \)
- \( \sqrt{x} \)
- \( \sqrt{36} \cdot \sqrt{x} \)

The correct equivalent expression for \( x^{1/3} \) is:

\[
\text{root index 3 Start Root x End Root}
\]

So, the answer is \( \sqrt[3]{x} \).