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

x−−√3

3 answers

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 to our expression:

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

Now, we need to express \( x^{32} \) in terms of the provided response options.

We can rewrite \( x^{32} \) as follows: \[ x^{32} = (x^2)^{16} = \sqrt[3]{x^{48}} = \sqrt{x^{32}} \]

However, we don't see an exact expression for \( x^{32} \) in the options given. Instead, let's verify each option to see which one can be simplified to \( x^{32} \):

  1. Option 1: \( \frac{1}{\sqrt[3]{\sqrt{x}}} \) does not correspond.
  2. Option 2: \( \sqrt{x} \) corresponds to \( x^{1/2} \).
  3. Option 3: \( \sqrt[36]{x} \) which corresponds to \( x^{1/36} \).
  4. Option 4: \( \sqrt[3]{x} \) corresponds to \( x^{1/3} \).

None of these match \( x^{32} \), indicating that this question likely seeks the equivalent of \( x^{32} \) under a different format, possibly not among the standard responses.

Thus, none of the provided expressions is equivalent to \( x^{16} \cdot x^{16} \). Could you please check if there might be additional expressions or a rephrasing needed?

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

x−−√3

To simplify the expression \( x^{16} \cdot x^{16} \), we apply the properties of exponents:

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

Now we need to find which of the provided options is equivalent to \( x^{32} \).

Let's look at each option:

  1. Option 1: \( \frac{1}{\sqrt[3]{x}} \)

    • This is \( x^{-1/3} \).

  2. Option 2: \( \sqrt{x} \)

    • This is \( x^{1/2} \).

  3. Option 3: \( \sqrt[36]{x} \)

    • This is \( x^{1/36} \).

  4. Option 4: \( \sqrt[3]{x} \)

    • This is \( x^{1/3} \).

None of these options equals \( x^{32} \) directly. However, we can find an expression for \( x^{32} \) in radical terms:

  • \( x^{32} \) can be expressed as:
    • \( (x^2)^{16} = \sqrt[16]{x^2} \)
    • But this approach isn’t directly available in the choices provided.

To simplify further, if we were to express \( x^{32} \) in terms of square roots, we'd use:

\[ x^{32} = \left(x^{16}\right)^2 = \left(\sqrt{x^{32}}\right)^2 \]

None of the provided choices equate this to \( x^{32} \). It's possible that the options given do not have a correct equivalent for \( x^{32} \). Please verify that the expressions match your expectations or provide additional context to the question.