Question

Using the properties of exponents, which expression is equivalent to x16⋅x16 ?(1 point) Responses x−−√36 root index 36 Start Root x End Root x−−√3 root index 3 Start Root x End Root 1x√3 Start Fraction 1 over root index 3 Start Root x End Root End Fraction x−−√

Answer 1

Using the properties of exponents, which expression is equivalent to x16⋅x16 ?(1 point) Responses
root index 36 Start Root x End Root x
root index 3 Start Root x End Root 1x
Start Fraction 1 over root index 3
Start Root x End Root End Fraction x−−√

Answer 2

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

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

In this case:

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

Now we need to check which of the given options is equivalent to \( x^{32} \):

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

None of these expressions simplify to \( x^{32} \). However, we can determine which expression can be evaluated further in terms of roots or exponents.

Given the expressions provided, there doesn't seem to be a direct equivalent to \( x^{32} \) among them, as all of them involve roots or fractions where the exponents would not sum to 32.

If \( x^{32} \) were to be expressed as a root, it would be \( \sqrt[n]{x^{32}} \), for some \( n \), but this option is not present.

Which specific equivalent is being sought in your responses? Could you clarify if we should derive it from the options given?

Answer 3

Let's evaluate \( x^{16} \cdot x^{16} \):

Using the property of exponents:
\[
x^a \cdot x^b = x^{a+b}
\]
we have:
\[
x^{16} \cdot x^{16} = x^{16 + 16} = x^{32}
\]

Next, we will convert \( x^{32} \) into a form that matches one of the provided responses:

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

Now, we need to express \( x^{32} \) in root format:
\[
x^{32} = \sqrt[1]{x^{32}} = \sqrt[1]{x^{32}} \text{ or } x^{32} = \sqrt[1]{x^{32}}.
\]

Let's express \( x^{32} \) using roots:
\[
x^{32} = \sqrt[1]{x^{32}} \quad \text{(not in given options)}.
\]

None of the options \( \sqrt[36]{x} \), \( \sqrt[3]{x} \), \( \frac{1}{\sqrt[3]{x}} \), or \( \sqrt{x} \) are equivalent to \( x^{32} \).

### Conclusion:
The expression \( x^{16} \cdot x^{16} = x^{32} \) does not match any of the provided options. Please verify if there's a mistake in the options or if any additional choices exist.