Question

Using the properties of exponents, which expression is equivalent to x^1/6 * x^1/6

sqrt of x
\root(3)(x)
\root((1)/(3))(x)
\root(36)(x)

Answer 1

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

Here, we have:

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

Next, we can express \( x^{1/3} \) in radical form. The exponent \( \frac{1}{3} \) corresponds to the cube root:

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

Now, looking at the choices given:

1. \(\sqrt{x}\) = \(x^{1/2}\) (not equivalent)
2. \(\root{3}{x}\) = \(x^{1/3}\) (equivalent)
3. \(\root{(1/3)}{x}\) = this is ambiguous and not typically used (not equivalent)
4. \(\root{36}{x}\) = \(x^{1/36}\) (not equivalent)

The correct equivalent expression for \( x^{1/6} \cdot x^{1/6} \) is:

\(\root{3}{x}\) (option 2).