Question

Using the properties of exponents, which expression is equivalent to x1/6⋅x1/6 ?(1 point)
Responses

x−−√
Start Root x End Root

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

x−−√36

Answer 1

To simplify the expression \( x^{1/6} \cdot x^{1/6} \), we can use the properties of exponents. When multiplying two expressions with the same base, we add their exponents:

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

Now, \( x^{1/3} \) can be expressed as the cube root of \( x \):

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

Now comparing with the given options:

1. \( \sqrt{x} \) (This corresponds to \( x^{1/2} \))
2. \( \sqrt[3]{x} \) (This corresponds to \( x^{1/3} \))
3. \( \frac{1}{\sqrt[3]{x}} \) (This corresponds to \( x^{-1/3} \))
4. \( \sqrt[3]{36} \) (This doesn't match anything we simplified)

The correct equivalent expression is:

**\( \sqrt[3]{x} \)**.