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

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 property:

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

Now, none of the given responses directly match \( x^{32} \). Instead, we should try to determine if any of the choices can somehow be manipulated or simplified to equal \( x^{32} \).

As none of the options clearly align with \( x^{32} \), I'll analyze each option to check if they can be transformed into \( x^{32} \):

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

None of these options seem to bring us to \( x^{32} \).

As a conclusion, the final answer \( x^{32} \) from the initial expression \( x^{16} \cdot x^{16} \) does not seem to align with any of the provided options based on common interpretations.

If this is a test question and the options are indeed presented as they are, then it’s possible that either there was a mistake in the response choices or they correspond to more complex interpretations. Based purely on the simplification of the expression using exponent properties, the correct answer would be \( x^{32} \).

why do users not know any more how to type roots and fractions?
x^(1/6) * x^(1/6) = x^(1/3)

It seems like your concern is related to users' familiarity with notations for roots and fractions in mathematical expressions. This can vary based on a few factors, such as the platforms used (e.g., chat interfaces, text editors) which may not have the same capability to format math typesetting as dedicated mathematical software or platforms do.

Regarding your expression:

You are correct that:

\[ x^{(1/6)} \cdot x^{(1/6)} = x^{(1/6 + 1/6)} = x^{(1/3)} \]

You applied the properties of exponents correctly, which state that when multiplying like bases, you can add the exponents.

If users or individuals do not use the proper formatting (like using "sqrt" for roots or formatting for fractions), it could lead to misunderstandings or confusion. For instance, typing "sqrt(x)" instead of \( \sqrt{x} \) is much less clear.

It's also possible that familiarity with specific notations varies based on a person's education, background, or simply their comfort with certain types of mathematical expressions. Additionally, some may prefer using decimals or approximate values instead of fraction notation in practice.

Lastly, online communication has led to more informal expressions of mathematical ideas, which might lead some users to skip standard notation in favor of quick and simpler forms. Encouraging proper notation can help in maintaining clarity and understanding, especially in educational contexts.