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

Question

?(1 point)
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
Start Fraction 1 over root index 3 Start Root x End Root End Fraction

Answer 1

To simplify the expression \( x^{16} \cdot x^{16} \), we use the properties of exponents, specifically that when multiplying two expressions with the same base, we add the exponents:

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

Now, we need to determine which of the given options is equivalent to \( x^{32} \).

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

None of these expressions are equivalent to \( x^{32} \) as all of them represent much smaller powers of \( x \).

Therefore, the expression \( x^{16} \cdot x^{16} \) simplifies to \( x^{32} \), which is not equivalent to any of the options provided.