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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−−√

3 answers

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?

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−−√

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.