Question

Using the properties of exponents, which expression is equivalent to x16⋅x16?
Responses

x−−√
Start Root x End Root

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

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

1x√3

Answer 1

To simplify the expression \( x^{16} \cdot x^{16} \), we can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \).

Applying this property:

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

Now, let's check the response options to find an equivalent expression:

1. **\( \sqrt{x} \)** - This is \( x^{1/2} \), which is not equivalent to \( x^{32} \).
2. **\( \sqrt[3]{x} \)** - This is \( x^{1/3} \), which is not equivalent to \( x^{32} \).
3. **\( \sqrt[36]{x} \)** - This is \( x^{1/36} \), which is not equivalent to \( x^{32} \).
4. **\( \frac{1}{\sqrt[3]{x}} \)** - This is \( x^{-1/3} \), which is not equivalent to \( x^{32} \).

None of the given responses are equivalent to \( x^{32} \). If you are looking for the correct expression, it should simply be \( x^{32} \) or in a different form like \( \sqrt[32]{x^{32}} \) that would still equal \( x \).

If you need to check any other options not listed here, let me know!