Question

Using the properties of exponents, which expression is equivalent to x16⋅x16 ?(1 point)
Responses

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

x−−√
Start Root x End Root

x−−√36
root index 36 Start Root x End Root

1x√3

Answer 1

To simplify the expression \( x^{16} \cdot x^{16} \) using the properties of exponents, we can apply the product of powers rule, which states that when multiplying two exponents with the same base, you add the exponents:

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

In this case, we have:

\[
x^{16} \cdot x^{16} = x^{16 + 16} = x^{32}
\]

Now, none of the provided options are directly equivalent to \( x^{32} \), so let's see if any of them can be simplified to \( x^{32} \).

1. \( \sqrt[3]{x} = x^{1/3} \)
2. \( \sqrt{x} = x^{1/2} \)
3. \( \sqrt[36]{x} = x^{1/36} \)
4. \( \frac{1}{\sqrt[3]{x}} = x^{-1/3} \)

None of these representations are equivalent to \( x^{32} \). Therefore, among the choices provided, none are equivalent to \( x^{16} \cdot x^{16} \) or \( x^{32} \).