Question

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

1x√3
Start Fraction 1 over root index 3 Start Root x End Root End Fraction

x−−√
Start Root x End Root

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

x−−√36

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 analyze the response options you provided:

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

We need to see which of these expressions can be simplified to \( x^{32} \).

1. **Option 1**:
\[ \frac{1}{\sqrt[3]{\sqrt{x}}} = \frac{1}{(x^{1/2})^{1/3}} = \frac{1}{x^{1/6}} = x^{-1/6} \]

2. **Option 2**:
\[ \sqrt{x} = x^{1/2} \]

3. **Option 3**:
\[ \sqrt[3]{\sqrt{x}} = (x^{1/2})^{1/3} = x^{1/6} \]

4. **Option 4**:
\[ \sqrt{x^{36}} = x^{36/2} = x^{18} \]

None of these options simplify to \( x^{32} \). Therefore, among the given options, none are equivalent to \( x^{32} \).

If the question is asking for which expression represents something related to \( x^{32} \) or a mathematical transformation of it, that information isn't clear in your provided expressions. Please ensure the options correspond to the exact simplification of \( x^{16} \cdot x^{16} = x^{32} \).